(CBC) in which customers choose the most preferred product among available
choices. Such CBC models used together with the analysis tools such as Logit
(McFadden, 1977) have been successful in different NPF applications (Meeran et
al., 2017).

In the third approach, mathematical/formal models known as growth or diffusion
curves (see §2.3.18 and §2.3.19) have been used successfully to do NPF (Hu et
al., 2019). The non-availability of past data is mitigated by growth curves by
capturing the generic pattern of the demand growth of a class of products,
which could be defined by a limited number of parameters such as saturation
level, inflexion point, etc. For a new product a growth curve can be
constituted from well-estimated parameters using analogous products, market
intelligence or regression methods. Most extensively used family of growth
curves for NPF has started with Bass model (Bass, 1969) that has been extended
extensively (Easingwood et al., 1983; Simon and Sebastian, 1987; Bass et al.,
2001; Islam and Meade, 2000; Peres et al., 2010). A recent applications of NPF
focused on consumer electronic goods using analogous products (Goodwin et al.,
2013a).

####  3.2.7 Spare parts forecasting989898This subsection was written by
Mohamed Zied Babai.

Spare parts are ubiquitous in modern societies. Their demand arises whenever a
component fails or requires replacement. Demand for spare parts is typically
intermittent, which means that it can be forecasted using the plethora of
parametric and non-parametric methods presented in §2.8. In addition to the
intermittence of demand, spare parts have two additional characteristics that
make them different from Work-In-Progress and final products, namely: (i) they
are generated by maintenance policies and part breakdowns, and (ii) they are
subject to obsolescence (Bacchetti and Saccani, 2012; Kennedy et al., 2002).

The majority of forecasting methods do not link the demand to the generating
factors, which are often related to maintenance activities. The demand for
spare parts originates from the replacement of parts in the installed base of
machines (i.e., the location and number of products in use), either
preventively or upon breakdown of the part (Kim et al., 2017). Fortuin (1984)
claims that using installed base information to forecast the spare part demand
can lead to stock reductions of up to 25%. An overview of the literature that
deals with spare parts forecasting with installed base information is given by
Van der Auweraer and Boute (2019). Spare parts demand can be driven by the
result of maintenance inspections and, in this case, a maintenance-based
forecasting model should then be considered to deal with this issue. Such
forecasting models include the Delay Time (DT) model analysed in Wang and
Syntetos (2011). Using the fitted values of the distribution parameters of a
data set related to a hospital pumps, Wang and Syntetos (2011) have shown that
when the failure and fault arriving characteristics of the items can be
captured, it is recommended to use the DT model to forecast the spare part
demand with a higher forecast accuracy. However, when such information is not
available, then time series forecasting methods, such as those presented in
§2.8.1, are recommended. The maintenance based forecasting is further
discussed in §3.2.8.

Given the life cycle of products, spare parts are associated with a risk of
obsolescence. Molenaers et al. (2012) discussed a case study where 54% of the
parts stocked at a large petrochemical company had seen no demand for the last
5 years. Hinton (1999) reported that the US Department of Defence was holding
60% excess of spare parts, with 18% of the parts (with a total value of $1.5
billion) having no demand at all. To take into account the issue of
obsolescence in spare parts demand forecasting, Teunter et al. (2011) have
proposed the TSB method, which deals with linearly decreasing demand and
sudden obsolescence cases. By means of an empirical investigation based on the
individual demand histories of 8000 spare parts SKUs from the automotive
industry and the Royal Air Force (RAF, UK), Babai et al. (2014) have
demonstrated the high forecast accuracy and inventory performance of the TSB
method. Other variants of the Croston’s method developed to deal with the risk
of obsolescence in forecasting spare parts demand include the Hyperbolic-
Exponential Smoothing method proposed by Prestwich et al. (2014) and the
modified Croston’s method developed by Babai et al. (2019).

####  3.2.8 Predictive maintenance999999This subsection was written by Juan
Ramón Trapero Arenas.

A common classification of industrial maintenance includes three types of
maintenance (Montero Jimenez et al., 2020). Corrective maintenance refers to
maintenance actions that occur after the failure of a component. Preventive
maintenance consists of maintenance actions that are triggered after a
scheduled number of units as cycles, kilometers, flights, etc. To schedule the
fixed time between two preventive maintenance actions, the Weibull
distribution is commonly used (Baptista et al., 2018). The drawbacks of
preventive maintenance are related to the replacement of components that still
have a remaining useful life; therefore, early interventions imply a waste of
resources and too late actions could imply catastrophic failures.
Additionally, the preventive intervention itself could be a source of failures
too. Finally, predictive maintenance (PdM) complements the previous ones and,
essentially, uses predictive tools to determine when actions are necessary
(Carvalho et al., 2019). Within this predictive maintenance group, other terms
are usually found in the literature as Condition-Based Maintenance and
Prognostic and Health Management, (Montero Jimenez et al., 2020).

The role of forecasting in industrial maintenance is of paramount importance.
One application is to forecast spare parts (see §3.2.7), whose demands are
typically intermittent, usually required to carry out corrective and
preventive maintenances (Wang and Syntetos, 2011; Van der Auweraer and Boute,
2019). On the other hand, it is crucial for PdM the forecast of the remaining
useful time, which is the useful life left on an asset at a particular time of
operation (Si et al., 2011). This work will be focused on the latter, which is
usually found under the prognostic stage (Jardine et al., 2006).

The typology of forecasting techniques employed is very ample. Montero Jimenez
et al. (2020) classify them in three groups: physics-based models, knowledge-
based models, and data-driven models. Physics-based models require high skills
on the underlying physics of the application. Knowledge-based models are based
on facts or cases collected over the years of operation and maintenance.
Although, they are useful for diagnostics and provide explicative results, its
performance on prognostics is more limited. In this sense, data-driven models
are gaining popularity for the development of computational power, data
acquisition, and big data platforms. In this case, data coming from vibration
analysis, lubricant analysis, thermography, ultrasound, etc. are usually
employed. Here, well-known forecasting models as VARIMAX/GARCH (see also §2.3)
are successfully used (García et al., 2010; Cheng et al., 2012; Gomez Munoz et
al., 2014; Baptista et al., 2018). State Space models based on the Kalman
Filter are also employed (Pedregal and Carmen Carnero, 2006; Pedregal et al.,
2009, and §2.3.6). Recently, given the irruption of the Industry 4.0, physical
and digital systems are getting more integrated and Machine
Learning/Artificial Intelligence are drawing the attention of practitioners
and academics alike (Carvalho et al., 2019). In that same reference, it is
found that the most frequently used Machine Learning methods in PdM
applications were Random Forest, Artificial Neural Networks, Support Vector
Machines and K-means.

####  3.2.9 Reverse logistics100100100This subsection was written by Aris A.
Syntetos.

As logistics and supply chain operations rely upon accurate demand forecasts
(see also §3.2.2), reverse logistics and closed loop supply chain operations
rely upon accurate forecasts of returned items. Such items (usually referred
as cores) can be anything from reusable shipping or product containers to used
laptops, mobile phones or car engines. If some (re)manufacturing activity is
involved in supply chains, it is both demand and returned items forecasts that
are needed since it is net demand requirements (demand – returns) that drive
remanufacturing operations.

Forecasting methods that are known to work well when applied to demand
forecasting, such as SES for example (see §2.3.1), do not perform well when
applied to time-series of returns because they assume returns to be a process
independent of sales. There are some cases when this independence might hold,
such as when a recycler receives items sold by various companies and supply
chains (Goltsos and Syntetos, 2020). In these cases, simple methods like SES
applied on the time series of returns might prove sufficient. Typically
though, returns are strongly correlated with past sales and the installed base
(number of products with customers). After all, there cannot be a product
return if a product has not first been sold. This lagged relationship between
sales and returns is key to the effective characterisation of the returns
process.

Despite the increasing importance of circular economy and research on closed
loop supply chains, returns forecasting has not received sufficient attention
in the academic literature (notable contributions in this area include Goh and
Varaprasad, 1986; Toktay et al., 2000; Toktay, 2003; de Brito and van der
Laan, 2009; Clottey et al., 2012). The seminal work by Kelle and Silver (1989)
offers a useful framework to forecasting that is based on the degree of
available information about the relationship between demand and returns.
Product level (PL) information consists of the time series of sales and
returns, alongside information on the time each product spends with a
customer. The question then is how to derive this time to return distribution.
This can be done through managerial experience, by investigating the
correlation of the demand and the returns time series, or by serialising and
tracking a subset (sample) of items. Past sales can then be used in
conjunction with this distribution to create forecasts of returns. Serial
number level (SL) information, is more detailed and consists of the time
matching of an individual unit item’s issues and returns and thus exactly the
time each individual unit, on a serial number basis, spent with the customer.
Serialisation allows for a complete characterisation of the time to return
distribution. Very importantly, it also enables tracking exactly how many
items previously sold remain with customers, providing time series of
unreturned past sales. Unreturned past sales can then be extrapolated –- along
with a time to return distribution -– to create forecasts of returns.

Goltsos et al. (2019) offered empirical evidence in the area of returns
forecasting by analysing a serialised data set from a remanufacturing company
in North Wales. They found the Beta probability distribution to best fit
times-to-return. Their research suggests that serialisation is something
worthwhile pursuing for low volume products, especially if they are expensive.
This makes a lot of sense from an investment perspective, since the relevant
serial numbers are very few. However, they also provided evidence that such
benefits expand in the case of high volume items. Importantly, the benefits of
serialisation not only enable the implementation of the more complex SL
method, but also the accurate characterisation of the returns process, thus
also benefiting the PL method (which has been shown to be very robust).

###  3.3 Economics and finance

####  3.3.1 Macroeconomic survey expectations101101101This subsection was
written by Michael P. Clements.

Macroeconomic survey expectations allow tests of theories of how agents form
their expectations. Expectations play a central role in modern macroeconomic
research (Gali, 2008). Survey expectations have been used to test theories of
expectations formation for the last 50 years. Initially the Livingston survey
data on inflationary expectations was used to test extrapolative or adaptive
hypothesis, but the focus soon turned to testing whether expectations are
formed rationally (see Turnovsky and Wachter, 1972, for an early
contribution). According to (Muth, 1961, p.316), rational expectations is the
hypothesis that: ‘expectations, since they are informed predictions of future
events, are essentially the same as the predictions of the relevant economic
theory.’ This assumes all agents have access to all relevant information.
Instead, one can test whether agents make efficient use of the information
they possess. This is the notion of forecast efficiency (Mincer and Zarnowitz,
1969), and can be tested by regressing the outturns on a constant and the
forecasts of those outturns. Under forecast efficiency, the constant should be
zero and the coefficient on the forecasts should be one. When the slope
coefficient is not equal to one, the forecast errors will be systematically
related to information available at the forecast origin, namely, the
forecasts, and cannot be optimal. The exchange between Figlewski and Wachtel
(1981, 1983) and Dietrich and Joines (1983) clarifies the role of partial
information in testing forecast efficiency (that is, full information is not
necessary), and shows that the use of the aggregate or consensus forecast in
the individual realisation-forecast regression outlined above will give rise
to a slope parameter less than one when forecasters are efficient but possess
partial information. Zarnowitz (1985), Keane and Runkle (1990) and Bonham and
Cohen (2001) consider pooling across individuals in the realisation-forecast
regression, and the role of correlated shocks across individuals.

Recently, researchers considered why forecasters might not possess full-
information, stressing informational rigidities: sticky information (see,
inter alia, Mankiw and Reis, 2002; Mankiw et al., 2003), and noisy information
(see, inter alia, Woodford, 2002; Sims, 2003). Coibion and Gorodnichenko
(2012, 2015) test these models using aggregate quantities, such as mean errors
and revisions.

Forecaster behaviour can be characterised by the response to new information
(see also §2.11.1). Over or under-reaction would constitute inefficiency.
Broer and Kohlhas (2018) and Bordalo et al. (2018) find that agents over-
react, generating a negative correlation between their forecast revision and
error. The forecast is revised by more than is warranted by the new
information (over-confidence regarding the value of the new information).
Bordalo et al. (2018) explain the over-reaction with a model of ‘diagnostic’
expectations, whereas Fuhrer (2018) finds ‘intrinsic inflation persistence’:
individuals under-react to new information, smoothing their responses to news.

The empirical evidence is often equivocal, and might reflect: the vintage of
data assumed for the outturns; whether allowance is made for ‘instabilities’
such as alternating over- and under-prediction (Rossi and Sekhposyan, 2016)
and the assumption of squared-error loss (see, for example, Patton and
Timmermann, 2007; Clements, 2014b).

Research has also focused on the histogram forecasts produced by a number of
macro-surveys. Density forecast evaluation techniques such as the probability
integral transform102102102See, for example, Rosenblatt (1952), Shephard
(1994), Kim et al. (1998), Diebold et al. (1998) and Berkowitz (2001). have
been applied to histogram forecasts, and survey histograms have been compared
to benchmark forecasts (see, for example, Bao et al., 2007; Hall and Mitchell,
2009; Clements, 2018). Research has also considered uncertainty measures based
on the histograms Clements (2014a). §2.12.4 and §2.12.5 also discuss the
evaluation and reliability of probabilistic forecasts.

Engelberg et al. (2009) and Clements (2009, 2010) considered the consistency
between the point predictions and histogram forecasts. Reporting practices
such as ‘rounding’ have also been considered (Binder, 2017; Manski and
Molinari, 2010; Clements, 2011).

Clements (2019) reviews macroeconomic survey expectations.

####  3.3.2 Forecasting GDP and inflation103103103This subsection was written
by Alessia Paccagnini.

As soon as Bayesian estimation of DSGEs became popular, these models have been
employed in forecasting horseraces to predict the key macro variables, for
example, Gross Domestic Product (GDP) and inflation, as discussed in Del Negro
and Schorfheide (2013). The forecasting performance is evaluated using rolling
or recursive (expanded) prediction windows (for a discussion, see Cardani et
al., 2015). DSGEs are usually estimated using revised data, but several
studies propose better results estimating the models using real-time data
(see, for example, Del Negro and Schorfheide, 2013; Wolters, 2015; Kolasa and
Rubaszek, 2015b; Cardani et al., 2019).

The current DSGE model forecasting compares DSGE models to competitors (see
§2.3.15 for an introduction to DSGE models). Among them, we can include the
Random Walk (the naive model which assumes a stochastic trend), the Bayesian
VAR models (Minnesota Prior à la Doan et al., 1984; and Large Bayesian VAR à
la Bańbura et al., 2010), the Hybrid-Models (the DSGE-VAR à la Del Negro and
Schorfheide, 2004; and the DSGE-Factor Augmented VAR à la Consolo et al.,
2009), and the institutional forecasts (Greenbook, Survey Professional
Forecasts, and the Blue Chip, as illustrated in Edge and Gürkaynak, 2010).

Table 2 summarises the current DSGE forecasting literature mainly for the US
and Euro Area and provided by estimating medium-scale models. As general
findings, DSGEs can outperform other competitors, with the exception for the
Hybrid-Models, in the medium and long-run to forecast GDP and inflation. In
particular, Smets and Wouters (2007) was the first empirical evidence of how
DSGEs can be competitive with forecasts from Bayesian VARs, convincing
researchers and policymakers in adopting DSGEs for prediction evaluations. As
discussed in Del Negro and Schorfheide (2013), the accuracy of DSGE forecasts
depends on how the model is able to capture low-frequency trends in the data.
To explain the macro-finance linkages during the Great Recession, the Smets
and Wouters model was also compared to other DSGE specifications including the
financial sector. For example, Del Negro and Schorfheide (2013), Kolasa and
Rubaszek (2015a), Galvão et al. (2016), and Cardani et al. (2019) provide
forecasting performance for DSGEs with financial frictions. This strand of the
literature shows how this feature can improve the baseline Smets and Wouters
predictions for the business cycle, in particular during the recent Great
Recession.

However, the Hybrid-Models always outperform the DSGEs thanks to the
combination of the theory-based model (DSGE) and the statistical
representation (VAR or Factor Augmented VAR), as illustrated by Del Negro and
Schorfheide (2004) and Consolo et al. (2009).

Moreover, several studies discuss how prediction performance could depend on
the parameters’ estimation. Kolasa and Rubaszek (2015b) suggest that updating
DSGE model parameters only once a year is enough to have accurate and
efficient predictions about the main macro variables.

Table 2: Alternative Competitors to DSGE Models Competitor |  Reference
---|---
Hybrid Models |  US: Del Negro and Schorfheide (2004), Consolo et al. (2009)
Random Walk |  US: Gürkaynak et al. (2013), Euro Area: Warne et al. (2010), Smets et al. (2014)
Bayesian VAR |  US: Smets and Wouters (2007), Gürkaynak et al. (2013), Wolters (2015), Bekiros and Paccagnini (2014), Bekiros and Paccagnini (2015a), Bekiros and Paccagnini (2015b), Euro Area: Warne et al. (2010)
Time-Varying VAR and Markov-Switching |  US: Bekiros et al. (2016), Euro Area: Bekiros and Paccagnini (2016)
Institutional Forecasts |  US: Edge and Gürkaynak (2010), Kolasa et al. (2012), Del Negro and Schorfheide (2013), Wolters (2015)

####  3.3.3 Forecasting unemployment104104104This subsection was written by
Jennifer L. Castle.

Unemployment has significant implications at both the micro and macro levels,
influencing individual living standards, health and well-being, as well as
imposing direct costs on the economy. Given its importance, policy-makers put
unemployment at the heart of their economic plans, and as such require
accurate forecasts to feed into economic policy decisions. Unemployment is
described as a lagging indicator of the economy, with characteristics
including business cycles and persistence. Despite this, forecasting the
unemployment rate is difficult, because the data are highly non-stationary
with abrupt distributional shifts, but persistence within regimes. In this
section we focus on methods used to forecast the aggregate unemployment rate.

Unemployment is the outcome of supply and demand for labour, aggregated across
all prospective workers, with labour demand derived from demand for goods and
services. This implies a highly complex data generating process. Empirical
forecasting models tend to simplify this relationship, with two approaches
dominating the literature. The first is based on the Phillips (1958) curve
capturing a non-linear relationship between nominal wage inflation and the
unemployment rate, or the relation between unemployment and output described
as Okun’s 1962 Law. The second uses the time-series properties of the data to
produce statistical forecasts, such as univariate linear models (for example,
ARIMA or unobserved component models; see §2.3.4 and §2.3.6), multivariate
linear models (for example, VARMA or CVAR; see §2.3.9), various threshold
autoregressive models (see §2.3.13), Markov Switching models (see §2.3.12) and
Artificial Neural Networks (see §2.7.8).

The empirical literature is inconclusive as to the ‘best’ forecasting models
for unemployment, which varies by country, time period and forecast horizon.
There is some evidence that non-linear statistical models tend to outperform
within business cycle contractions or expansions, but perform worse across
business cycles (see, for example, Montgomery et al., 1998; Rothman, 1998;
Koop and Potter, 1999), whereas Proietti (2003) finds that linear models
characterised by higher persistence perform significantly better. Evidence of
non-linearities is found by Peel and Speight (2000), Milas and Rothman (2008)
and Johnes (1999), and Gil-Alana (2001) finds evidence of long-memory.
Barnichon and Garda (2016) applies a flow approach to unemployment forecasting
and finds improvements, as does Smith (2011).

One approach that does yield accurate forecasts is to use a measure of
profitability as the explanatory variable, assuming that unemployment will
fall when hiring is profitable. Hendry (2001) proxies profitability ($\pi$) by
the gap between the real interest rate (reflecting costs) and the real growth
rate (reflecting the demand side), such that the unemployment rate rises when
the real interest rate exceeds the real growth rate, and vice versa:

$\pi_{t}=\left(R_{L}-\Delta p-\Delta y\right)_{t}$

where $R_{L}$ is the long-term interest rate, $\Delta p$ is a measure of
inflation and $\Delta y$ is a measure of output growth. This is then embedded
within a dynamic equilibrium correction model, using impulse indicator
saturation (IIS: Hendry et al., 2008b; Johansen and Nielsen, 2009) and step
indicator saturation (SIS: Castle et al., 2015a) to capture outliers, breaks
and regime shifts, as well as allowing for any non-linearities using Taylor
expansions for the regressors. The resulting forecasts perform well over the
business cycle relative to alternative statistical models (also see Hendry,
2015, and Castle, Hendry, and Martinez, 2020c).

Forecasts from models of unemployment could be improved with either better
economic theories of aggregate unemployment,105105105There are many relevant
theories based on microfoundations, including search and matching, loss of
skills, efficiency wages, and insider-outsider models, see Layard et al.
(1991) for a summary. or more general empirical models that tackle stochastic
trends, breaks, dynamics, non-linearities and interdependence,106106106See
Hendry and Doornik (2014) for an approach to jointly tackling all of these
issues. or better still, both. The COVID-19 pandemic and subsequent lockdown
policies highlight just how important forecasts of unemployment are (Castle et
al., 2021).

####  3.3.4 Forecasting productivity107107107This subsection was written by
Andrew B. Martinez.

The growth of labour productivity, measured by the percent change in output
per hours worked, has varied dramatically over the last 260 years. In the UK
it ranged from -5.8% at the onset of the 1920 Depression to just over 7% in
1971; see panel A in figure 6. Productivity growth is very volatile and has
undergone large historical shifts with productivity growth averaging around 1%
between 1800-1950 followed by an increase in the average annual growth to 3%
between 1950-1975. Since the mid-1970’s productivity growth has gradually
declined in many developed economies; see panel B of figure 6. In the decade
since 2009, 2% annual productivity growth was an upper bound for most G7
countries.

Figure 6: Productivity Growth (Output per total hours worked). Sources: Bank
of England and Penn World Table Version 10.0.

The most common approach for forecasting productivity is to estimate the trend
growth in productivity using aggregate data. For example, Gordon (2003)
considers three separate approaches for calculating trend labor productivity
in the United States based on (i) average historical growth rates outside of
the business cycle, (ii) filtering the data using the HP filter (Hodrick and
Prescott, 1997), and (iii) filtering the data using the Kalman filter (see
Kalman, 1960). The Office for Budget Responsibility (OBR) in the UK and the
Congressional Budget Office (CBO) in the US follow similar approaches for
generating its forecasts of productivity based on average historical growth
rates as well as judgments about factors that may cause productivity to
deviate from its historical trend in the short-term.108108108See
https://obr.uk/forecasts-in-depth/the-economy-forecast/potential-output-and-
the-output-gap. (Accessed: 2020-09-05) Alternative approaches include
forecasting aggregate productivity using disaggregated firm-level data (see
Bartelsman et al., 2011; Bartelsman and Wolf, 2014, and §2.10.1) and using
time-series models (see Žmuk et al., 2018, and §2.3.4).

In the last few decades there have been several attempts to test for time-
varying trends in productivity and to allow for them. However, the focus of
these approaches has been primarily on the United States (Hansen, 2001;
Roberts, 2001), which saw a sharp rise in productivity growth in the 1990’s
that was not mirrored in other countries (Basu et al., 2003). Test for shifts
in productivity growth rates in other advanced economies did not find evidence
of a changes in productivity growth until well after the financial crisis in
2007 (Benati, 2007; Turner and Boulhol, 2011; Glocker and Wegmüller, 2018).

A more recent approach by Martinez et al. (2021) allows for a time-varying
long-run trend in UK productivity. They show that are able to broadly
replicate the OBR’s forecasts using a quasi-transformed autoregressive model
with one lag, a constant, and a trend. The estimated long-run trend is just
over 2% per year through 2007 Q4 which is consistent with the OBR’s
assumptions about the long-run growth rate of productivity (OBR, 2019).
However, it is possible to dramatically improve upon OBR’s forecasts in real-
time by allowing for the long-term trend forecast to adjust based on more
recent historical patterns. By taking a local average of the last four years
of growth rates, Martinez et al. (2021) generate productivity forecasts whose
RMSE is on average more than 75% smaller than OBR’s forecasts extending five-
years-ahead and is 84% smaller at the longest forecast horizon.

####  3.3.5 Fiscal forecasting for government budget surveillance109109109This
subsection was written by Diego J. Pedregal.

Recent economic recessions have led to a renewed interest in fiscal
forecasting, mainly for deficit and debt surveillance. This was certainly true
in the case of the 2008 recession, and looks to become even more important in
the current economic crisis brought on by the COVID-19 pandemic. This is
particularly important in Europe, where countries are subject to strong fiscal
monitoring mechanisms. Two main themes can be detected in the fiscal
forecasting literature (Leal et al., 2008). First, investigate the properties
of forecasts in terms of bias, efficiency and accuracy. Second, check the
adequacy of forecasting procedures.

The first topic has its own interest for long, mainly restricted to
international institutions (Artis and Marcellino, 2001). Part of the
literature, however, argue that fiscal forecasts are politically biased,
mainly because there is usually no clear distinction between political targets
and rigorous forecasts (Strauch et al., 2004; Frankel and Schreger, 2013). In
this sense, the availability of forecasts from independent sources is of great
value (Jonung and Larch, 2006). But it is not as easy as saying that
independent forecasters would improve forecasts due to the absence of
political bias, because forecasting accuracy is compromised by complexities of
data, country-specific factors, outliers, changes in the definition of fiscal
variables, etc. Very often some of these issues are known by the staff of
organisations in charge of making the official statistics and forecasts long
before the general public, and some information never leaves such
institutions. So this insider information is actually a valuable asset to
improve forecasting accuracy (Leal et al., 2008).

As for the second issue, namely the accuracy of forecasting methods, the
literature can be divided into two parts, one based on macroeconomic models
with specific fiscal modules that allows to analyse the effects of fiscal
policy on macro variables and vice versa (see Favero and Marcellino (2005) and
references therein), and the other based on pure forecasting methods and
comparisons among them. This last stream of research basically resembles
closely what is seen in other forecasting areas: (i) there is no single method
outperforming the rest generally, (ii) judgmental forecasting is especially
important due to data problems (see §2.11), and (iii) combination of methods
tends to outperform individual ones, see Leal et al. (2008) and §2.6.

Part of the recent literature focused on the generation of very short-term
public finance monitoring systems using models that combine annual information
with intra-annual fiscal data (Pedregal and Pérez, 2010) by time aggregation
techniques (see §2.10.2), often set up in a SS framework (see §2.3.6). The
idea is to produce global annual end-of-year forecasts of budgetary variables
based on the most frequently available fiscal indicators, so that changes
throughout the year in the indicators can be used as early warnings to infer
the changes in the annual forecasts and deviations from fiscal targets
(Pedregal et al., 2014).

The level of disaggregation of the indicator variables are established
according to the information available and the particular objectives. The
simplest options are the accrual National Accounts annual or quarterly fiscal
balances running on their cash monthly counterparts. A somewhat more complex
version is the previous one with all the variables broken down into revenues
and expenditures. Other disaggregation schemes have been applied, namely by
region, by administrative level (regional, municipal, social security, etc.),
or by items within revenue and/or expenditure (VAT, income taxes, etc. Paredes
et al., 2014; Asimakopoulos et al., 2020).

Unfortunately, what is missing is a comprehensive and transparent forecasting
system, independent of Member States, capable of producing consistent
forecasts over time and across countries. This is certainly a challenge that
no one has yet dared to take up.

####  3.3.6 Interest rate prediction110110110This subsection was written by
Massimo Guidolin & Manuela Pedio.

The (spot) rate on a (riskless) bond represents the ex-ante return (yield) to
maturity which equates its market price to a theoretical valuation. Modelling
and predicting default-free, short-term interest rates are crucial tasks in
asset pricing and risk management. Indeed, the value of interest
rate–sensitive securities depends on the value of the riskless rate. Besides,
the short interest rate is a fundamental ingredient in the formulation and
transmission of the monetary policy (see, for example, §2.3.15). However, many
popular models of the short rate (for instance, continuous time, diffusion
models) fail to deliver accurate out-of-sample forecasts. Their poor
predictive performance may depend on the fact that the stochastic behaviour of
short interest rates may be time-varying (for instance, it may depend on the
business cycle and on the stance of monetary policy).

Notably, the presence of nonlinearities in the conditional mean and variance
of the short-term yield influences the behaviour of the entire term structure
of spot rates implicit in riskless bond prices. For instance, the level of the
short-term rate directly affects the slope of the yield curve. More generally,
nonlinear rate dynamics imply a nonlinear equilibrium relationship between
short and long-term yields. Accordingly, recent research has reported that
dynamic econometric models with regime shifts in parameters, such as Markov
switching (MS; see §2.3.12) and threshold models (see §2.3.13), are useful at
forecasting rates.

The usefulness of MS VAR models with term structure data had been established
since Hamilton (1988) and Garcia and Perron (1996): single-state, VARMA models
are overwhelmingly rejected in favour of multi-state models. Subsequently, a
literature has emerged that has documented that MS models are required to
successfully forecast the yield curve. Lanne and Saikkonen (2003) showed that
a mixture of autoregressions with two regimes improves the predictions of US
T-bill rates. Ang and Bekaert (2002) found support for MS dynamics in the
short-term rates for the US, the UK, and Germany. Cai (1994) developed a MS
ARCH model to examine volatility persistence, reflecting a concern that it may
be inflated by regimes. Gray (1996) generalised this attempt to MS GARCH and
reported improvements in pseudo out-of-sample predictions. Further advances in
the methods and applications of MS GARCH are in Haas et al. (2004) and Smith
(2002). A number of papers have also investigated the presence of regimes in
the typical factors (level, slope, and convexity) that characterise the no-
arbitrage dynamics of the term structure, showing the predictive benefits of
incorporating MS (see, for example, Guidolin and Pedio, 2019; Hevia et al.,
2015).

Alternatively, a few studies have tried to capture the time-varying, nonlinear
dynamics of interest rates using threshold models. As discussed by Pai and
Pedersen (1999), threshold models have an advantage compared to MS ones: the
regimes are not determined by an unobserved latent variable, thus fostering
interpretability. In most of the applications to interest rates, the regimes
are determined by the lagged level of the short rate itself, in a self-
exciting fashion. For instance, Pfann et al. (1996) explored nonlinear
dynamics of the US short-term interest rate using a (self-exciting) threshold
autoregressive model augmented by conditional heteroskedasticity (namely, a
TAR-GARCH model) and found strong evidence of the presence of two regimes.
More recently, also Gospodinov (2005) used a TAR-GARCH to predict the short-
term rate and showed that this model can capture some well-documented features
of the data, such as high persistence and conditional heteroskedasticity.

Another advantage of nonlinear models is that they can reproduce the empirical
puzzles that plague the expectations hypothesis of interest rates (EH),
according to which it is a weighted average of short-term rates to drive
longer-term rates (see, for example, Bansal et al., 2004; Dai et al., 2007).
For instance, while Bekaert et al. (2001) show single-state VARs cannot
generate distributions consistent with the EH, Guidolin and Timmermann (2009)
find that the optimal combinations of lagged short and forward rates depend on
regimes so that the EH holds only in some states.

As widely documented (see, for instance, Guidolin and Thornton, 2018), the
predictable component in mean rates is hardly significant. As a result, the
random walk remains a hard benchmark to outperform as far as the prediction of
the mean is concerned. However, density forecasts reflect all moments and the
models that capture the dynamics of higher-order moments tend to perform best.
MS models appear at the forefront of a class of non-linear models that produce
accurate density predictions (see, for example, Hong et al., 2004; Maheu and
Yang, 2016). Alternatively, Pfann et al. (1996) and more recently Dellaportas
et al. (2007) estimated TAR models to also forecast conditional higher order
moments and all report reasonable accuracy.

Finally, a literature has strived to fit rates not only under the physical
measure, i.e., in time series, but to predict rates when MS enters the pricing
kernel, the fundamental pricing operator. A few papers have assumed that
regimes represent a new risk factor (see, for instance, Dai and Singleton,
2003). This literature reports that MS models lead to a range of shapes for
nominal and real term structures (see, for instance, Veronesi and Yared,
1999). Often the model specifications that are not rejected by formal tests
include regimes (Ang et al., 2008; Bansal and Zhou, 2002).

To conclude, it is worthwhile noting that, while threshold models are more
interpretable, MS remain a more popular alternative for the prediction of
interest rates. This is mainly due to the fact that statistical inference for
threshold regime switching models poses some challenges, because the
likelihood function is discontinuous with respect to the threshold parameters.

####  3.3.7 House price forecasting111111111This subsection was written by
Alisa Yusupova.

The boom and bust in housing markets in the early and mid 2000s and its
decisive role in the Great Recession has generated a vast interest in the
dynamics of house prices and emphasised the importance of accurately
forecasting property price movements during turbulent times. International
organisations, central banks and research institutes have become increasingly
engaged in monitoring the property price developments across the
world.112112112For instance, the International Monetary Fund recently
established the Global Housing Watch, the Globalisation and Monetary Policy
Institute of the Federal Reserve Bank of Dallas initiated a project on
monitoring international property price dynamics, and the UK Housing
Observatory initiated a similar project for the UK national and regional
housing markets. At the same time, a substantial empirical literature has
developed that deals with predicting future house price movements (for a
comprehensive survey see Ghysels et al., 2013). Although this literature
concentrates almost entirely on the US (see, for example, Rapach and Strauss,
2009; Bork and Møller, 2015), there are many other countries, such as the UK,
where house price forecastability is of prime importance. Similarly to the US,
in the UK, housing activities account for a large fraction of GDP and of
households’ expenditures; real estate property comprises a significant
component of private wealth and mortgage debt constitutes a main liability of
households (Office for National Statistics, 2019).

The appropriate forecasting model has to reflect the dynamics of the specific
real estate market and take into account its particular characteristics. In
the UK, for instance, there is a substantial empirical literature that
documents the existence of strong spatial linkages between regional markets,
whereby the house price shocks emanating from southern regions of the country,
and in particular Greater London, have a tendency to spread out and affect
neighbouring regions with a time lag (see, for example, Cook and Thomas, 2003;
Holly et al., 2010; Antonakakis et al., 2018, inter alia); see also §2.3.10 on
forecasting functional data.

Recent evidence also suggests that the relationship between real estate
valuations and conditioning macro and financial variables displayed a complex
of time-varying patterns over the previous decades (Aizenman and Jinjarak,
2013). Hence, predictive methods that do not allow for time-variation in both
predictors and their marginal effects may not be able to capture the complex
house price dynamics in the UK (see Yusupova et al., 2019, for a comparison of
forecasting accuracy of a battery of static and dynamic econometric methods).

An important recent trend is to attempt to incorporate information from novel
data sources (such as newspaper articles, social media, etc.) in forecasting
models as a measure of expectations and perceptions of economic agents (see
also §2.9.3). It has been shown that changes in uncertainty about house prices
impact on housing investment and real estate construction decisions
(Cunningham, 2006; Banks et al., 2015; Oh and Yoon, 2020), and thus
incorporating a measure of uncertainty in the forecasting model can improve
the forecastability of real estate prices. For instance in the UK, the House
Price Uncertainty (HPU) index (Yusupova et al., 2020), constructed using the
methodology outlined in Baker et al. (2016),113113113For a comparison of
alternative text-based measures of economic uncertainty see Kalamara et al.
(2020). was found to be important in predicting property price inflation ahead
of the house price collapse of the third quarter of 2008 and during the bust
phase (Yusupova et al., 2019). Along with capturing the two recent recessions
(in the early 1990s and middle 2000s) this index also reflects the uncertainly
related to the EU Referendum, Brexit negotiations and COVID-19 pandemic.

####  3.3.8 Exchange rate forecasting114114114This subsection was written by
Michał Rubaszek.

Exchange rates have long fascinated and puzzled researchers in international
finance. The reason is that following the seminal paper of Meese and Rogoff
(1983), the common wisdom is that macroeconomic models cannot outperform the
random walk in exchange rate forecasting (see Rossi, 2013, for a survey). This
view is difficult to reconcile with the strong belief that exchange rates are
driven by fundamentals, such as relative productivity, external imbalances,
terms of trade, fiscal policy or interest rate disparity (MacDonald, 1998; Lee
et al., 2013; Couharde et al., 2018). These two contradicting assertions by
the academic literature is referred to as “exchange rate disconnect puzzle”.

The literature provides several explanations for this puzzle. First, it can be
related to the forecast estimation error (see §2.5.2). The studies in which
models are estimated with a large panels of data (Mark and Sul, 2001; Engel et
al., 2008; Ince, 2014), long time series (Lothian and Taylor, 1996) or
calibrated (Ca’ Zorzi and Rubaszek, 2020) deliver positive results on exchange
rate forecastability. Second, there is ample evidence that the adjustment of
exchange rates to equilibrium is non-linear (Taylor and Peel, 2000; Curran and
Velic, 2019), which might diminish the out-of-sample performance of
macroeconomic models (Kilian and Taylor, 2003; Lopez-Suarez and Rodriguez-
Lopez, 2011). Third, few economists argue that the role of macroeconomic
fundamentals may be varying over time and this should be accounted for in a
forecasting setting (Byrne et al., 2016; Beckmann and Schussler, 2016).

The dominant part of the exchange rate forecasting literature investigates
which macroeconomic model performs best out-of-sample. The initial studies
explored the role of monetary fundamentals to find that these models deliver
inaccurate short-term and not so bad long-term predictions in comparison to
the random walk (Meese and Rogoff, 1983; Mark, 1995). In a comprehensive study
from mid-2000s, Cheung et al. (2005) showed that neither monetary, uncovered
interest parity (UIP) nor behavioural equilibrium exchange rate (BEER) model
are able to outperform the no-change forecast. A step forward was made by
Molodtsova and Papell (2009), who proposed a model combining the UIP and
Taylor rule equations and showed that it delivers competitive exchange rate
forecasts. This result, however, has not been confirmed by more recent studies
(Cheung et al., 2019; Engel et al., 2019). In turn, Ca’ Zorzi and Rubaszek
(2020) argue that a simple method assuming gradual adjustment of the exchange
rate towards the level implied by the Purchasing Power Parity (PPP) performs
well over shorter as well as longer horizon. This result is consistent with
the results of Ca’ Zorzi et al. (2017) and Eichenbaum et al. (2017), who
showed that exchange rates are predictable within a general equilibrium DSGE
framework (see §2.3.15), which encompasses an adjustment of the exchange rate
to a PPP equilibrium. Finally, Ca’ Zorzi et al. (2020) discuss how extending
the PPP framework for other fundamentals within the BEER framework is not
helping in exchange rate forecasting. Overall, at the current juncture it
might be claimed that “exchange rate disconnect puzzle” is still puzzling,
with some evidence that methods based on PPP and controlling the estimation
forecast error can deliver more accurate forecast than the random walk
benchmark. A way forward to account for macroeconomic variables in exchange
rate forecasting could be to use variable selection methods that allow to
control for the estimation error (see §2.5.3).

####  3.3.9 Financial time series forecasting with range-based volatility
models115115115This subsection was written by Piotr Fiszeder.

The range-based (RB) volatility models is a general term for the models
constructed with high and low prices, and most often with their difference
i.e., the price range. A short review and classification of such models is
contained in §2.3.14. From practical point of view, it is important that low
and high prices are almost always available with daily closing prices for
financial series. The price range (or its logarithm) is a significantly more
efficient estimator of volatility than the estimator based on closing prices
(Alizadeh et al., 2002). Similarly the co-range (the covariance based on price
ranges) is a significantly more efficient estimator of the covariance of
returns than the estimator based on closing prices (Brunetti and Lildholdt,
2002). For these reasons models based on the price range and the co-range
better describe variances and covariances of financial returns than the ones
based on closing prices.

Forecasts of volatility from simple models like moving average, EWMA, AR, ARMA
based on the RB variance estimators are more accurate than the forecasts from
the same models based on squared returns of closing prices (Vipul and Jacob,
2007; Rajvanshi, 2015). Forecasts of volatility from the AR model based on the
Parkinson estimator are more precise even than the forecasts from the standard
GARCH models (see §2.3.11) based on closing prices (Li and Hong, 2011).

In plenty of studies it was shown that forecasts of volatility of financial
returns from the univariate RB models are more accurate than the forecasts
from standard GARCH models based on closing prices (see, for example, Mapa,
2003 for the GARCH-PARK-R model; Chou, 2005 for the CARR model; Fiszeder, 2005
for the GARCH-TR model; Brandt and Jones, 2006 for the REGARCH model; Chen et
al., 2008 for the TARR model; Lin et al., 2012 for the STARR model; Fiszeder
and Perczak, 2016 for the GARCH model estimated with low, high and closing
prices during crisis periods; Molnár, 2016 for the RGARCH model).

The use of daily low and high prices in the multivariate volatility models
leads to more accurate forecasts of covariance or covariance matrix of
financial returns than the forecasts from the models based on closing prices
(see, for example, Chou et al., 2009 for the RB DCC model; Harris and Yilmaz,
2010 for the hybrid EWMA model; Fiszeder, 2018 for the BEKK-HL model; Fiszeder
and Fałdziński, 2019 for the co-range DCC model; Fiszeder et al., 2019 for the
DCC-RGARCH model).

The RB models were used in many financial applications. They lead for example
to more precise forecasts of value-at-risk measures in comparison to the
application of only closing prices (see, for example, Chen et al., 2012 for
the threshold CAViaR model; Asai and Brugal, 2013 for the HVAR model; Fiszeder
et al., 2019 for the DCC-RGARCH model; Meng and Taylor, 2020 for scoring
functions). The application of the multivariate RB models provides also the
increase in the efficiency of hedging strategies (see, for example, Chou et
al., 2009 for the RB DCC model; Harris and Yilmaz, 2010 for the hybrid EWMA
model; Su and Wu, 2014 for the RB-MS-DCC model). Moreover, the RB volatility
models have more significant economic value than the return-based ones in the
portfolio construction (Chou and Liu, 2010 for the RB DCC model; Wu and Liang,
2011 for the RB-copula model). Some studies show that based on the forecasts
from the volatility models with low and high prices it is possible to
construct profitable investment strategies (He et al., 2010 for the VECM
model; Kumar, 2015 for the CARRS model).

####  3.3.10 Copula forecasting with multivariate dependent financial times
series116116116This subsection was written by Feng Li.

In this section, we focus on the practical advances on jointly forecasting
multivariate financial time series with copulas. In the copula framework (see
§2.4.3), because marginal models and copula models are separable, point
forecasts are straightforward with marginal models, but dependence information
is ignored. A joint probabilistic forecast with copulas involves both
estimations of the copula distribution and marginal models.

In financial time series, an emerging interest is to model and forecast the
asymmetric dependence. A typical asymmetric dependence phenomenon is that two
stock returns exhibit greater correlation during market downturns than market
upturns. Patton (2006a) employs the asymmetric dependence between exchange
rates with a time-varying copula construction with AR and GARCH margins. A
similar study for measuring financial contagion with copulas allows the
parameters of the copula to change with the states of the variance to identify
shifts in the dependence structure in times of crisis (Rodriguez, 2007).

In stock forecasting, Almeida and Czado (2012) employ a stochastic copula
autoregressive model to model DJI and Nasdaq, and the dependence at the time
is modelled by a real-valued latent variable, which corresponds to the Fisher
transformation of Kendall’s $\tau$. Li and Kang (2018) use a covariate-
dependent copula framework to forecast the time varying dependence that
improves both the probabilistic forecasting performance and the forecasting
interpretability. Liquidity risk is another focus in finance. Weiß and Supper
(2013) forecast three types of liquidity-adjusted intraday Value-at-Risk
(L-IVaR) with a vine copula structure. The liquidity-adjusted intraday VaR is
based on simulated portfolio values, and the results are compared with the
realised portfolio profits and losses.

In macroeconomic forecasting, most existing reduced-form models for
multivariate time series produce symmetric forecast densities. Gaussian
copulas with skew Student’s-_t_ margins depict asymmetries in the predictive
distributions of GDP growth and inflation (Smith and Vahey, 2016). Real-time
macroeconomic variables are forecasted with heteroscedastic inversion copulas
(Smith and Maneesoonthorn, 2018) that allow for asymmetry in the density
forecasts, and both serial and cross-sectional dependence could be captured by
the copula function (Loaiza-Maya and Smith, 2020).

Copulas are also widely used to detect and forecast default correlation, which
is a random variable called _time-until-default_ to denote the survival time
of each defaultable entity or financial instrument (Li, 2000). Then copulas
are used in modelling the dependent defaults (Li, 2000), forecasting credit
risk (Bielecki and Rutkowski, 2013), and credit derivatives market forecasting
(Schönbucher, 2003). A much large volume of literature is available for this
specific area. See the aforementioned references therein. For particular
applications in credit default swap (CDS) and default risk forecasting, see Oh
and Patton (2018) and Li and He (2019) respectively.

In energy economics, Aloui et al. (2013) employ the time-varying copula
approach, where the marginal models are from ARMA($p$,$q$)–GARCH(1,1) to
investigate the conditional dependence between the Brent crude oil price and
stock markets in the Central and Eastern European transition economies. Bessa
et al. (2012) propose a time-adaptive quantile-copula where the copula density
is estimated with a kernel density forecast method. The method is applied to
wind power probabilistic forecasting (see also §3.4.6) and shows its
advantages for both system operators and wind power producers. Vine copula
models are also used to forecast wind power farms’ uncertainty in power system
operation scheduling. Wang et al. (2017) shows vine copulas have advantages of
providing reliable and sharp forecast intervals, especially in the case with
limited observations available.

####  3.3.11 Financial forecasting with neural networks117117117This
subsection was written by Georgios Sermpinis.

Neural Networks (NNs; see §2.7.8) are capable of successfully modelling non-
stationary and non-linear series. This property has made them one of the most
popular (if not the most) non-linear specification used by practitioners and
academics in Finance. For example, 89% of European banks use NNs to their
operations (European Banking Federation, 2019) while 25.4% of the NNs
applications in total is in Finance (Wong et al., 1995).

The first applications of NNs in Finance and currently the most widespread, is
in financial trading. In the mid-80s when computational power became cheaper
and more accessible, hedge fund managers started to experiment with NNs in
trading. Their initial success led to even more practitioners to apply NNs and
nowadays 67% of hedge fund managers use NNs to generate trading ideas
(BarclayHedge, 2018). A broad measure of the success of NNs in financial
trading is provided by the Eurekahedge AI Hedge Fund
Index118118118https://www.eurekahedge.com/Indices/IndexView/Eurekahedge/683/Eurekahedge-
AI-Hedge-fund-Index (Accessed: 2020-09-01) where it is noteworthy the 13.02%
annualised return of the selected AI hedge funds over the last 10 years.

In academia, financial trading with NNs is the focus of numerous papers.
Notable applications of NNs in trading financial series were provided by
Kaastra and Boyd (1996), Tenti (1996), Panda and Narasimhan (2007), Zhang and
Ming (2008), and Dunis et al. (2010). The aim of these studies is to forecast
the sign or the return of financial trading series and based on these
forecasts to generate profitable trading strategies. These studies are closely
related to the ones presented in §3.3.13 but the focus is now in
profitability. The second major field of applications of NNs in Finance is in
derivatives pricing and financial risk management. The growth of the financial
industry and the provided financial services have made NNs and other machine
learning algorithms a necessity for tasks such as fraud detection, information
extraction and credit risk assessment (Buchanan, 2019). In derivatives
pricing, NNs try to fill the limitations of the Black-Scholes model and are
being used in options pricing and hedging. In academia notable applications of
NNs in risk management are provided by Locarek-Junge and Prinzler (1998) and
Liu (2005) and in derivatives by Bennell and Sutcliffe (2004) and Psaradellis
and Sermpinis (2016).

As discussed before, financial series due to their non-linear nature and their
wide applications in practice seems the perfect forecasting data set for
researchers that want to test their NN topologies. As a result, there are
thousands of forecasting papers in the field of NNs in financial forecasting.
However, caution is needed in interpretation of their results. NNs are
sensitive to the choice of their hyperparameters. For a simple MLP, a
practitioner needs to set (among others) the number and type of inputs, the
number of hidden nodes, the momentum, the learning rate, the number of epochs
and the batch size. This complexity in NN modelling leads inadvertently to the
data snooping bias (see also §2.12.6). In other words, a researcher that
experiments long enough with the parameters of a NN topology can have
excellent in-sample and out-of-sample results for a series. However, this does
not mean that the results of his NN can be generalised. This issue has led the
related literature to be stained by studies cannot be extended in different
samples.

####  3.3.12 Forecasting returns to investment style119119119This subsection
was written by Ross Hollyman.

Investment style or factor portfolios are constructed from constituent
securities on the basis of a variety of a-priori observable characteristics,
thought to affect future returns. For example a ‘Momentum’ portfolio might be
constructed with positive (‘long’) exposures to stocks with positive trailing
12-month returns, and negative (‘short’) exposure to stocks with negative
trailing 12-month returns (for full background and context, see, for example
Bernstein, 1995; Haugen, 2010).120120120The website of Kenneth French is an
excellent source of data on investment style factor data and research.
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Explanations as to why such characteristics seem to predict returns fall in to
two main camps: firstly that the returns represent a risk premium, earned by
the investor in return for taking on some kind of (undiversifiable) risk, and
secondly that such returns are the result of behavioural biases on the part of
investors. In practice, both explanations are likely to drive style returns to
a greater or lesser extent. Several such strategies have generated reasonably
consistent positive risk-adjusted returns over many decades, but as with many
financial return series, return volatility is large relative to the mean, and
there can be periods of months or even years when returns deviate
significantly from their long-run averages. The idea of timing exposure to
styles is therefore at least superficially attractive, although the
feasibility of doing so is a matter of some debate (Arnott et al., 2016;
Asness, 2016; Bender et al., 2018). Overconfidence in timing ability has a
direct cost in terms of trading frictions and opportunity cost in terms of
potential expected returns and diversification forgone.

A number of authors write on the general topic of style timing (recent
examples include Hodges et al., 2017; Dichtl et al., 2019), and several
forecasting methodologies have been suggested, falling in to three main camps:

  1. 1.

Serial Correlation: Perhaps the most promising approach is exploiting serial
correlation in style returns. Tarun and Bryan (2019) and Babu et al. (2020)
outline two such approaches and Ehsani and Linnainmaa (2020) explore the
relationship between momentum in factor portfolios and momentum in underlying
stock returns. As with valuation spreads mentioned below, there is a risk that
using momentum signals to time exposure to momentum factor portfolios risks
unwittingly compounding exposure. A related strand of research relates (own)
factor volatility to future returns, in particular for momentum factors
(Barroso, 2015; Daniel and Moskowitz, 2016).

  2. 2.

Valuation Spreads: Using value signals (aggregated from individual stock value
exposures) to time exposure to various fundamental-based strategies is a
popular and intuitively appealing approach (Asness, 2016); however evidence of
value added from doing so is mixed, and the technique seems to compound risk
exposure to value factors.

  3. 3.

Economic & Financial Conditions: Polk et al. (2020) explore how economic and
financial conditions affect style returns (an idea that dates back at least to
Bernstein (1995) and references therein).

Style returns exhibit distinctly non-normal distributions. On a univariate
basis, most styles display returns which are highly negatively skewed and
demonstrate significant kurtosis. The long-run low correlation between
investment styles is often put forward as a benefit of style-based strategies,
but more careful analysis reveals that non-normality extends to the co-
movements of investment style returns; factors exhibit significant tail
dependence. Christoffersen and Langlois (2013) explores this issue, also
giving details of the skew and kurtosis of weekly style returns. These
features of the data mean that focusing solely on forecasting the mean may not
be sufficient, and building distributional forecasts becomes important for
proper risk management. Jondeau (2007) writes extensively on modelling non-
gaussian distributions.

####  3.3.13 Forecasting stock returns121121121This subsection was written by
David E. Rapach.

Theory and intuition suggest a plethora of potentially relevant predictors of
stock returns. Financial statement data (e.g., Chan and Genovese, 2001; Yan
and Zheng, 2017) provide a wealth of information, and variables relating to
liquidity, price trends, and sentiment, among numerous other concepts, have
been used extensively by academics and practitioners alike to predict stock
returns. The era of big data further increases the data available for
forecasting returns. When forecasting with large numbers of predictors,
conventional ordinary least squares (OLS) estimation is highly susceptible to
overfitting, which is exacerbated by the substantial noise in stock return
data (reflecting the intrinsically large unpredictable component in returns);
see §2.7.11.

Over the last decade or so, researchers have explored methods for forecasting
returns with large numbers of predictors. Principal component regression
extracts the first few principal components (or factors) from the set of
predictors; the factors then serve as predictors in a low-dimensional
predictive regression, which is estimated via OLS (see §2.7.1). Intuitively,
the factors combine the information in the individual predictors to reduce the
dimension of the regression, which helps to guard against overfitting.
Ludvigson and Ng (2007) find that a few factors extracted from hundreds of
macroeconomic and financial variables improve out-of-sample forecasts of the
US market return. Kelly and Pruitt (2013) and Huang et al. (2015b) use partial
least squares (Wold, 1966) to construct target-relevant factors from a cross
section of valuation ratios and a variety of sentiment measures, respectively,
to improve market return forecasts.

Since Bates and Granger (1969), it has been known that combinations of
individual forecasts often perform better than the individual forecasts
themselves (Timmermann, 2006, and §2.6.1). Rapach et al. (2010) show that
forecast combination can significantly improve out-of-sample market return
forecasts. They first construct return forecasts via individual univariate
predictive regressions based on numerous popular predictors from the
literature (Goyal and Welch, 2008). They then compute a simple combination
forecast by taking the average of the individual forecasts. Rapach et al.
(2010) demonstrate that forecast combination exerts a strong shrinkage effect,
thereby helping to guard against overfitting.

An emerging literature uses machine-learning techniques to construct forecasts
of stock returns based on large sets of predictors. In an investigation of
lead-lag relationships among developed equity markets, Rapach et al. (2013)
appear to be the first to employ machine-learning tools to predict market
returns. They use the elastic net (ENet, Zou and Hastie, 2005), a
generalisation of the popular least absolute shrinkage and selection operator
(LASSO, Tibshirani, 1996). The LASSO and ENet employ penalised regression to
guard against overfitting in high-dimensional settings by shrinking the
parameter estimates toward zero. Chinco et al. (2019) use the LASSO to
forecast high-frequency (one-minute-ahead) individual stock returns and report
improvements in out-of-sample fit, while Rapach et al. (2019) use the LASSO to
improve monthly forecasts of industry returns.

Incorporating insights from Diebold and Shin (2019), Han et al. (2021) use the
LASSO to form combination forecasts of cross-sectional stock returns based on
a large number of firm characteristics from the cross-sectional literature
(e.g., Harvey et al., 2016; McLean and Pontiff, 2016; Hou et al., 2020),
extending the conventional OLS approach of Haugen and Baker (1996), Lewellen
(2015), and Green et al. (2017). Rapach and Zhou (2020) and Dong et al.
(forthcoming) use the ENet to compute combination forecasts of the market
return based on popular predictors from the time-series literature and
numerous anomalies from the cross-sectional literature, respectively.
Forecasting individual stock returns on the basis of firm characteristics in a
panel framework, Freyberger et al. (2020) and Gu et al. (2020) employ machine-
learning techniques – such as the nonparametric additive LASSO (Huang et al.,
2010), random forests (Breiman, 2001), and artificial neural networks – that
allow for nonlinear predictive relationships.

####  3.3.14 Forecasting crashes in stock markets122122122This subsection was
written by Philip Hans Franses.

Time series data on financial asset returns have special features. Returns
themselves are hard to forecast, while it seems that volatility of returns can
be predicted. Empirical distributions of asset returns show occasional
clusters of large positive and large negative returns. Large negative returns,
that is, crashes seem to occur more frequently than large positive returns.
Forecasting upcoming increases or decreases in volatility can be achieved by
using variants of the Autoregressive Conditional Heteroskedasticity (ARCH)
model (Engle, 1982; Bollerslev, 1986, and §2.3.11) or realized volatility
models (Taylor, 1986b). These models take (functions of) past volatility and
past returns as volatility predictors, although also other explanatory
variables can be incorporated in the regression.

An important challenge that remains is to predict crashes. Sornette (2003)
summarises potential causes for crashes and these are computer trading,
increased trading in derivatives, illiquidity, trade and budget deficits, and
especially, herding behaviour of investors. Yet, forecasting the exact timing
of crashes may seem impossible, but on the other hand, it may be possible to
forecast the probability that a crash may occur within a foreseeable future.
Given the herding behaviour, any model to use for prediction should include
some self-exciting behaviour. For that purpose, Aït-Sahalia et al. (2015)
propose mutually exciting jump processes, where jumps can excite new jumps,
also across assets or markets (see also Chavez-Demoulin et al., 2005). Another
successful approach is the Autoregressive Conditional Duration (ACD) model
(Engle and Russell, 1997, 1998), which refers to a time series model for
durations between (negative) events.

An alternative view on returns’ volatility and the potential occurrence of
crashes draws upon the earthquake literature (Ogata, 1978, 1988). The idea is
that tensions in and across tectonic plates build up, until an eruption, and
after that, tension starts to build up again until the next eruption. By
modelling the tension-building-up process using so-called Hawkes processes
(Hawkes, 1971, 2018; Hawkes and Oakes, 1974; Ozaki, 1979), one can exploit the
similarities between earthquakes and financial crashes (see also §2.8.4).
Gresnigt et al. (2015) take Hawkes processes to daily S&P 500 data and show
that it is possible to create reliable probability predictions of a crash
occurrence within the next five days. Gresnigt et al. (2017a, b) further
develop a specification strategy for any type of asset returns, and document
that there are spillovers across assets and markets.

Given investor behaviour, past crashes can ignite future crashes. Hawkes
processes are particularly useful to describe this feature and can usefully be
implemented to predict the probability of nearby crashes. By the way, these
processes can also be useful to predict social conflicts, as also there one
may discern earthquake-like patterns. van den Hengel and Franses (2020)
document their forecasting power for social conflicts in Africa.

###  3.4 Energy

####  3.4.1 Building energy consumption forecasting and
optimisation123123123This subsection was written by Christoph Bergmeir &
Evangelos Spiliotis.

In Europe, buildings account for 40% of total energy consumed and 36% of total
$\text{CO}_{2}$ emissions (Patti et al., 2016). Given that energy consumption
of buildings is expected to increase in the coming years, forecasting
electricity consumption becomes critical for improving energy management and
planning by supporting a large variety of optimisation procedures.

The main challenge in electricity consumption forecasting is that building
energy systems are complex in nature, with their behaviour depending on
various factors related to the type (e.g., residential, office, entertainment,
business, and industrial) and the end-uses (e.g., heating, cooling, hot water,
and lighting) of the building, its construction, its occupancy, the occupants’
behaviour and schedule, the efficiency of the installed equipment, and the
weather conditions (Zhao and Magoulès, 2012). Special events, holidays, and
calendar effects can also affect the behaviour of the systems and further
complicate the consumption patterns, especially when forecasting at hourly or
daily level (see §2.3.5). As a result, producing accurate forecasts typically
requires developing tailored, building-specific methods.

To deal with this task, the literature focuses on three main classes of
forecasting methods, namely engineering, statistical, and ML (Mat Daut et al.,
2017). Engineering methods, typically utilised through software tools such as
DOE-2, EnergyPlus, BLAST, and ESP-r, build on physical models that forecast
consumption through detailed equations which account for the particularities
of the building (Al-Homoud, 2001; Zhao and Magoulès, 2012; Foucquier et al.,
2013). Statistical methods usually involve linear regression (see §2.3.2),
ARIMA/ARIMAX (see §2.3.4), and exponential smoothing (see §2.3.1) models that
forecast consumption using past consumption data or additional explanatory
variables, such as weather or occupancy and calendar related information (Deb
et al., 2017). Finally, ML methods (see §2.7.10) typically involve neural
networks (see §2.7.8), support vector machines, and grey models that account
for multiple non-linear dependencies between the electricity consumed and the
factors influencing its value (Ahmad et al., 2014). Till present, the
literature has been inconclusive about which class of methods is the most
appropriate, with the conclusions drawn being subject to the examined building
type, data set used, forecasting horizon considered, and data frequency at
which the forecasts are produced (Wei et al., 2019). To mitigate this problem,
combinations of methods (see §2.6) and hybrids (see §2.7.13) have been
proposed, reporting encouraging results (Zhao and Magoulès, 2012; Mohandes et
al., 2019).

Other practical issues refer to data pre-processing. Electricity consumption
data is typically collected at high frequencies through smart meters and
therefore display noise and missing or extreme values due to monitoring issues
(see §2.7.11). As a result, verifying the quality of the input data through
diagnostics and data cleansing techniques (see §2.2.3 and §2.2.4), as well as
optimising the selected time frames, are important for improving forecasting
performance (Bourdeau et al., 2019). Similarly, it is critical to engineer
(see §2.2.5) and select (see §2.5.3) appropriate regressor variables which are
of high quality and possible to accurately predict to assist electricity
consumption forecasting. Finally, it must be carefully decided whether the
bottom-up, the top-down or a combination method (see §2.10.1) will be used for
producing reconciled forecasts at both building and end-use level (Kuster et
al., 2017), being also possibly mixed with temporal aggregation approaches
(Spiliotis et al., 2020c, but also §2.10.3).

Provided that accurate forecasts are available, effective energy optimisation
can take place at a building level or across blocks of buildings (see §3.4.10)
to reduce energy cost, improve network stability, and support efforts towards
a carbon-free future, by exploiting smart grid, internet of things (IoT), and
big data technologies along with recommendation systems (Marinakis et al.,
2020).

An example for a typical application in this area is the optimisation of
heating, ventilation, and air conditioning (HVAC) systems. The goal is to
minimise the energy use of the HVAC system under the constraints of
maintaining certain comfort levels in the building (Marinakis et al., 2017).
Though this is predominantly an optimisation exercise, forecasting comes in at
different points of the system as input into the optimisation, and many
problems in this space involve forecasting as a sub-problem, including energy
consumption forecasting, room occupancy forecasting, inside temperature
forecasting, (hyper-local) forecasts of outside temperature, and air pressure
forecasting for ventilation, among others. For instance, Krüger and Givoni
(2004) use a linear regression approach to predict inside temperatures in 3
houses in Brazil, and Ruano et al. (2006) propose the use of a neural network
to predict temperatures in a school building. Madaus et al. (2020) predict
hyper-local extreme heat events, combining global climate models and machine
learning models. Jing et al. (2018) predict air pressure to tackle the air
balancing problem in ventilation systems, using a support vector machine.

Predicting energy demand on a building/household level from smart meter data
is an important research topic not only for energy savings. In the building
space, Ahmad et al. (2017), Touzani et al. (2018), and Wang et al. (2018b)
predict building energy consumption of residential and commercial buildings
using decision tree-based algorithms (random forests and gradient boosted
trees) and neural networks to improve energy efficiency.

A recent trend in forecasting are global forecasting models, built across sets
of time series (Januschowski et al., 2020). (Recurrent) neural networks
(Bandara et al., 2020a; Hewamalage et al., 2021) are particularly suitable for
this type of processing due to their capabilities to deal with external inputs
and cold-start problems. Such capabilities are necessary if there are
different regimes in the simulations under which to predict, an example of
such a system for HVAC optimisation is presented by Godahewa et al. (2020).

More generally, many challenges in the space of building energy optimisation
are classical examples of so-called “predict then optimise” problems
(Demirovic et al., 2019; Elmachtoub and Grigas, 2017). Here, different
possible scenario predictions are obtained from different assumptions in the
form of input parameters. These input parameters are then optimised to achieve
a desired predicted outcome. As both prediction and optimisation are difficult
problems, they are usually treated separately (Elmachtoub and Grigas, 2017),
though there are now recent works where they are considered together (El
Balghiti et al., 2019; Demirovic et al., 2019), and this will certainly be an
interesting avenue for future research.

####  3.4.2 Electricity price forecasting124124124This subsection was written
by Luigi Grossi & Florian Ziel.

Forecasting electricity prices has various challenges that are highlighted in
the detailed review papers by Weron (2014). Even though there are economically
well motivated fundamental electricity price models, forecasting models based
on evaluating historic price data are the dominating the academic literature.
In recent years the focus on probabilistic forecasting grew rapidly, as they
are highly relevant for many applications in energy trading and risk
management, storage optimisation and predictive maintenance, (Ziel and
Steinert, 2018; Nowotarski and Weron, 2018). Electricity price data is highly
complex and is influenced by regulation. However, there is electricity trading
based on auctions and on continuous trading. Many markets like the US and
European markets organise day-ahead auctions for electricity prices, see
figure 7. Thus, we have to predict multivariate time series type data, (Ziel
and Weron, 2018). In contrast, intraday markets usually apply continuous
trading to manage short term variations due to changes in forecasts of
renewable energy and demand, and outages (Kiesel and Paraschiv, 2017).

Figure 7: Hourly German day-ahead electricity price data resulting from a two-
sided auction (top left) with corresponding 24 sale/supply and purchase/demand
curves for 24 May 2020 and highlighted curves for 17:00 (top right), power
generation and consumption time series (bottom left), and bid structure of 24
May 2020 17:00 (bottom right).

The key challenge in electricity price forecasting is to address all potential
characteristics of the considered market, most notably (some of them visible
in figure 7):

  1. 1.

(time-varying) autoregressive effects and (in)stationarity,

  2. 2.

calendar effects (daily, weekly and annual seasonality, holiday effects,
clock-change),

  3. 3.

(time-varying) volatility and higher moment effects,

  4. 4.

price spikes (positive and negative), and

  5. 5.

price clustering.

Some of those impacts can be explained by external inputs, that partially have
to be predicted in advance:

  1. 1.

load/demand/consumption (see §3.4.3),

  2. 2.

power generation, especially from the renewable energy sources (RES) of wind
and solar (see §3.4.6 and §3.4.8),

  3. 3.

relevant fuel prices (especially oil, coal, natural gas; see also §3.4.4),

  4. 4.

prices of emission allowances ($CO_{2}e$ costs),

  5. 5.

related power market prices (future, balancing and neighboring markets),

  6. 6.

availabilities of power plants and interconnectors,

  7. 7.

import/export flow related data, and

  8. 8.

weather effects (e.g. temperature due to cooling and heating and combined heat
and power (CHP) effects; see also §3.5.2).

Note that other weather effects might be relevant as well, but should be
covered from the fundamental point of view by the listed external inputs.
Obvious examples are wind speed for the wind power prediction, cloud cover for
the solar power production and illumination effects in the electricity
consumption.

Many of those external effects may be explained by standard economic theory
from fundamental electricity price models (Cludius et al., 2014; Kulakov and
Ziel, 2021). Even the simple supply stack model (merit order model), see
figure 8, explains many features and should be kept in mind when designing an
appropriate electricity price forecasting model.

Figure 8: Illustrative example of a supply stack model with inelastic demand
for different power plant types, roughly covering the situation in Germany
2020.

In recent years, statistical and machine learning methods gained a lot of
attraction in day-ahead electricity price forecasting. Even though the
majority of effects is linear there are specific non-linear dependencies that
can be explored by using non-linear models, especially neural networks (Dudek,
2016; Lago et al., 2018; Ugurlu et al., 2018; Marcjasz et al., 2019). Of
course this comes along with higher computational costs compared to linear
models. Fezzi and Mosetti (2020) illustrate that even simple linear models can
give highly accurate forecasts, if correctly calibrated. However, there seems
to be consensus that forecast combination is appropriate, particularly for
models that have different structures or different calibration window length
(Gaillard et al., 2016; Mirakyan et al., 2017; Hubicka et al., 2018).

Another increasing stream of electricity price forecasting models do not focus
on the electricity price itself, but the bid/sale/sell/supply and
ask/sell/purchase/demand curves of the underlying auctions (see figure 7, but
also Ziel and Steinert, 2016; Kulakov, 2020; Shah and Lisi, 2020; Mestre et
al., 2020). This sophisticated forecasting problem allows more insights for
trading applications and the capturing of price clusters.

In forecasting intraday markets the literature just started to grow quickly.
As the aforementioned market characteristics get less distinct if information
from day-ahead markets is taken into account appropriately. However, intraday
prices are usually more volatile and exhibit more stronger price spikes. Thus,
probabilistic forecasting is even more relevant (Janke and Steinke, 2019;
Narajewski and Ziel, 2020b). Recent studies showed that European markets are
close to weak-form efficiency. Thus naive point forecasting benchmarks perform
remarkably well (Oksuz and Ugurlu, 2019; Narajewski and Ziel, 2020a; Marcjasz
et al., 2020).

As pointed out above, predicting price spikes is particularly important in
practice, due to the high impact in decision making problems which occur
usually in extreme situations, see figure 8. Very high electricity prices are
usually observed in connection to high demand and low renewable energy
generation, sometimes together with sudden power plant failures. In contrast,
negative price spikes occur in oversupply situation, when there is low demand
but high penetration from wind and solar power. The presence of spikes is
explored in two main streams in literature: spike forecasting and prediction
of prices under normal regime through robust estimators.

Within the first set of papers, spikes are often modelled as one regime of
non-linear models for time series. This approach is followed by Mount et al.
(2006) focusing on regime-switching models with parameters driven by time-
varying variables and by Becker et al. (2008) who adopt Markov switching
models for spikes prediction. Christensen et al. (2009, 2012) suggest treating
and forecasting price spikes through Poisson autoregressive and discrete-time
processes, respectively. Herrera and González (2014) use a Hawkes model
combined with extreme events theory. Interregional links among different
electricity markets are used by Clements et al. (2015) and Manner et al.
(2016) to forecast electricity price spikes. A new procedure for the
simulation of electricity spikes has been recently proposed by Muniain and
Ziel (2020) utilising bivariate jump components in a mean reverting jump
diffusion model in the residuals.

The second stream of literature includes papers developing outlier detection
methods or robust estimators to improve the forecasting performance of the
models. Martínez-Álvarez et al. (2011) tackle the issue of outlier detection
and prediction defining “motifs”, that is patches of units preceding
observations marked as anomalous in a training set. Janczura et al. (2013)
focus on the detection and treatment of outliers in electricity prices. A very
similar approach, based on seasonal autoregressive models and outlier
filtering, is followed by Afanasyev and Fedorova (2019). Grossi and Nan (2019)
introduced a procedure for the robust statistical prediction of electricity
prices. The econometric framework is represented by the robust estimation of
non-linear SETAR processes. A similar approach has been followed by Wang et
al. (2020a) using an outlier-robust machine learning algorithm.

####  3.4.3 Load forecasting125125125This subsection was written by Ioannis
Panapakidis.

Load forecasting forms the basis where power system operation and planning
builds upon. Based on the time horizon of the forecasts, load forecasting can
be classified into very short-term (VSTLF), that refers to horizon from
several minutes ahead up to 1 hour, short-term (STLF), that spans from 1 hour
to 168 hours ahead, medium-term (MTLF), that spans from 168 hours to 1 year
ahead and finally, and long-term (LTLF) that concerns predictions from 1 year
to several years ahead. In VSTLF and STLF applications, the focus is on the
sub-hourly or hourly load. In MTLF and LTLF, the variables of interest can be
either monthly electricity peak load and total demand for energy.

Inputs differ in the various horizons. In VSTLF and STLF, apart from
meteorological data, day type identification codes are used. In LTLF,
macroeconomic data are used since total demand of energy is influenced by the
long-term modifications of the social and economic environments. Among the
horizons, special attention is placed at STLF. This is reflected by the
research momentum that have been placed in the load forecasting related
literature by other researchers (Hong and Fan, 2016). Processes like unit
commitment and optimal power flow rely on STLF (Saksornchai et al., 2005; Bo
and Li, 2012). Additionally, since competitive energy markets continually
evolve, STLF becomes vital for new market entities such as retailers,
aggregators, and prosumers for applications such as strategic bidding,
portfolio optimisation, and tariff design (Danti and Magnani, 2017; Ahmad et
al., 2019).

The models that can be found in the load forecasting related literature can in
general categorised into three types: time-series, machine learning, and
hybrid. Time-series models historically precede the others. Typical examples
of this family are ARMA, ARIMA, and others (see also §2.3.4). In the machine
learning models, the structure is usually determined via the training process.
NNs are commonly used. Once a NN is sufficiently trained, it can provide
forecasts for all types of forecasting horizons (Hippert et al., 2001). The
third category of models refers to the integration of two or more individual
forecasting approaches (see also see §2.7.13). For instance, a NN can be
combined with time series methods, with unsupervised machine learning
algorithms, data transformation, and with meta-heuristics algorithms (Bozkurt
et al., 2017; López et al., 2017; Lu et al., 2019; El-Hendawi and Wang, 2020).

Hybrid systems has been tested on validation data (through forecasting
competitions), power system aggregated load, and application oriented tasks.
Ma (2021) proposed an ensemble method based on a combination of various single
forecasters on GEFCom2012 forecasting competition data that outperformed
benchmark forecasters such as Theta method, NN, ARIMA, and others (see §2.12.7
for further discussions on forecasting competitions). For aggregated load
cases, researchers focus on different countries and energy markets. Zhang et
al. (2018) combined empirical mode decomposition (EMD), ARIMA, and wavelet
neural networks (WNN) optimised by the fruit fly algorithm on Australian
Market data and New York City data. Their approach was to separate the linear
and nonlinear components from original electricity load; ARIMA is used for
linear part while the WNN for the non-linear one.

Sideratos et al. (2020) proposed that a radial basis network that performs the
initial forecasting could serve as input to a convolutional neural network
that performs the final forecasting. The proposed model led to lower error
compared to the persistence model, NN, and SVM. Semero et al. (2020) focused
on the energy management of a microgrid located in China using EMD to
decompose the load, adaptive neuro-fuzzy inference system (ANFIS) for
forecasting and particle swarm intelligence (PSO) to optimize ANFIS
parameters. The results show that the proposed approach yielded superior
performance over four other methods. Faraji et al. (2020) proposed a hybrid
system for the scheduling of a prosumer microgrid in Iran. Various machine
learning algorithms provided load and weather forecasts. Through an
optimisation routine, the best individual forecast is selected. The hybrid
system displayed better accuracy from the sole application of the individual
forecasters.

####  3.4.4 Crude oil price forecasting126126126This subsection was written by
Xiaoqian Wang.

Crude oil, one of the leading energy resources, has contributed to over one-
third of the world’s energy consumption (Alvarez-Ramirez et al., 2003). The
fluctuations of the crude oil price have a significant impact on industries,
governments as well as individuals, with substantial up-and-downs of the crude
oil price bringing dramatic uncertainty for the economic and political
development (Kaboudan, 2001; Cunado and De Gracia, 2005). Thus, it is critical
to develop reliable methods to accurately forecast crude oil price movement,
so as to guard against the crude oil market extreme risks and improve
macroeconomic policy responses. However, the crude oil price movement suffers
from complex features such as nonlinearity, irregularities, dynamics and high
volatility (Kang et al., 2009; Alquist et al., 2013; Herrera et al., 2018, and
also §2.3.11), making the crude oil price forecasting still one of the most
challenging forecasting problems.

Some prior studies have suggested that the crude oil price movement is
inherently unpredictable, and it would be pointless and futile to attempt to
forecast future prices, see Miao et al. (2017) for a detailed summary. These
agnostics consider the naive no-change forecast as the best available forecast
value of future prices. In recent years, however, numerous studies result in
forecasts that are more accurate than naive no-change forecasts, making the
forecasting activities of crude oil prices promising (Alquist et al., 2013;
Baumeister et al., 2015).

Extensive research on crude oil price forecasting has focused predominantly on
the econometric models, such as VAR, ARCH-type, ARIMA, and Markov models (see,
for example, Mirmirani and Li, 2004; Agnolucci, 2009; Mohammadi and Su, 2010;
e Silva et al., 2010, and §2.3). In the forecasting literature, unit root
tests (see §2.3.4) are commonly applied to examine the stationarity of crude
oil prices prior to econometric modelling (Silvapulle and Moosa, 1999;
Serletis and Rangel-Ruiz, 2004; Rahman and Serletis, 2012). It is well-
documented that crude oil prices are driven by a large set of external
components, which are themselves hard to predict, including supply and demand
forces, stock market activities, oil-related events (e.g., war, weather
conditions), political factors, etc. In this context, researchers have
frequently considered structural models (see §2.3.9), which relate the oil
price movements to a set of economic factors. With so many econometric models,
is there an optimal one? Recently, de Albuquerquemello et al. (2018) proposed
a SETAR model, allowing for predictive regimes changing after a detected
threshold, and achieved performance improvements over six widely used
econometric models. Despite their high computational efficiency, the
econometric models are generally limited in the ability to nonlinear time
series modelling.

On the other hand, artificial intelligence and machine learning techniques,
such as belief networks, support vector machines (SVMs), recurrent neural
networks (RNNs), and extreme gradient boosting (XGBoost), provided powerful
solutions to recognise the nonlinear and irregular patterns of the crude oil
price movement with high automation (see, for example, Abramson and Finizza,
1991; Xie et al., 2006; Mingming and Jinliang, 2012; Gumus and Kiran, 2017).
However, challenges also exist in these techniques, such as computational cost
and overfitting. In addition, a large number of studies have increasingly
focused on the hybrid forecasting models (see also §2.7.13) based on
econometrics models and machine learning techniques (Jammazi and Aloui, 2012;
He et al., 2012; Baumeister and Kilian, 2015; Chiroma et al., 2015), achieving
improved performance. Notably, the vast majority of the literature has focused
primarily on the deterministic prediction, with much less attention paid to
the probabilistic prediction and uncertainty analysis. However, the high
volatility of crude oil prices makes probabilistic prediction more crucial to
reduce the risk in decision-making (Abramson and Finizza, 1995; Sun et al.,
2018).

####  3.4.5 Forecasting renewable energy technologies127127127This subsection
was written by Mariangela Guidolin.

The widespread adoption of renewable energy technologies, RETs, plays a
driving role in the transition to low-carbon energy systems, a key challenge
to face climate change and energy security problems. Forecasting the diffusion
of RETs is critical for planning a suitable energy agenda and setting
achievable targets in terms of electricity generation, although the available
time series are often very short and pose difficulties in modelling. According
to Rao and Kishore (2010), renewables’ typical characteristics such as low
load factor, need for energy storage, small size, high upfront costs create a
competitive disadvantage, while Meade and Islam (2015b) suggested that
renewable technologies are different from other industrial technological
innovations because, in the absence of focused support, they are not
convenient from a financial point of view. In this sense, policy measures and
incentive mechanisms, such as feed-in tariffs, have been used to stimulate the
market. As highlighted in Lee and Huh (2017b), forecasting RETs requires to
capture different socio-economic aspects, such as policy choices by
governments, carbon emissions, macroeconomic factors, economic and financial
development of a country, competitive strength of traditional energy
technologies.

The complex and uncertain environment concerning RETs deployment has been
faced in literature in several ways, in order to account for various
determinants of the transition process. A first stream of research employed a
bottom-up approach, where forecasts at a lower level are aggregated to higher
levels within the forecasting hierarchy. For instance Park et al. (2016)
realised a bottom-up analysis to study the optimum renewable energy portfolio,
while Lee and Huh (2017a) performed a three-step forecasting analysis, to
reflect the specificities of renewable sources, by using different forecasting
methods for each of the sources considered. A similar bottom-up perspective
was adopted in Zhang et al. (2020), by conducting a multi-region study, to
understand how multi-level learning may affect RETs dynamics, with the
regionalised model of investment and technological development, a general
equilibrium model linking a macro-economic growth with a bottom-up
engineering-based energy system model.

The relative newness of RETs has posed the challenge of forecasting with a
limited amount of data: in this perspective, several contributions applied the
‘Grey System’ theory, a popular methodology for dealing with systems with
partially unknown parameters (Kayacan et al., 2010). Grey prediction models
for RETs forecasting were proposed in Tsai et al. (2017), Lu (2019), Wu et al.
(2019), Moonchai and Chutsagulprom (2020), and Liu and Wu (2021).

Other studies developed forecasting procedures based on growth curves and
innovation diffusion models (see §2.3.18, §2.3.19, and §2.3.20): from the
seminal work by Marchetti and Nakicenovic (1979), contributions on the
diffusion of RETs were proposed by Guidolin and Mortarino (2010), Dalla Valle
and Furlan (2011), Meade and Islam (2015b), Lee and Huh (2017b), and Bunea et
al. (2020). Forecasting the diffusion of renewable energy technologies was
also considered within a competitive environment in Huh and Lee (2014),
Guidolin and Guseo (2016), Furlan and Mortarino (2018), and Guidolin and
Alpcan (2019).

####  3.4.6 Wind power forecasting128128128This subsection was written by
Jethro Browell.

Wind energy is a leading source of renewable energy, meeting 4.8% of global
electricity demand in 2018, more than twice that of solar energy (IEA, Paris,
2020). Kinetic energy in the wind is converted into electrical energy by wind
turbines according to a characteristic ‘power curve’. Power production is
proportion to the cube of the wind speed at low-to-moderate speeds, and above
this is constant at the turbine’s rated power. At very high or low wind speeds
no power is generated. Furthermore, the power curve is influenced by
additional factors including air density, icing, and degradation of the
turbine’s blades.

Forecasts of wind energy production are required from minutes to days-ahead to
inform the operation of wind farms, participation in energy markets and power
systems operations. However, the limited predictability of the weather (see
also §3.5.2) and the complexity of the power curve make this challenging. For
this reason, probabilistic forecasts are increasing used in practice (Bessa et
al., 2017). Their value for energy trading is clear (Pinson et al., 2007), but
quantifying value for power system operation is extremely complex. Wind power
forecasting may be considered a mature technology as many competing commercial
offerings exist, but research and development efforts to produce novel and
enhanced products is ongoing (see also §3.4.5).

Short-term forecasts (hours to days ahead) of wind power production are
generally produced by combining numerical weather predictions (NWP) with a
model of the wind turbine, farm or even regional power curve, depending on the
objective. The power curve may be modelled using physical information, e.g.
provided by the turbine manufacturer, in which case it is also necessary to
post-process NWP wind speeds to match the same height-above-ground as the
turbine’s rotor. More accurate forecasts can be produced by learning the NWP-
to-energy relationship from historic data when it is available. State-of-the-
art methods for producing wind power forecasts leverage large quantities of
NWP data to produce a single forecast (Andrade et al., 2017) and detailed
information about the target wind farm (Gilbert et al., 2020a). A number of
practical aspects may also need to be considered by users, such as maintenance
outages and requirements to reduce output for other reasons, such as noise
control or electricity network issues.

Very short-term forecast (minutes to a few hours ahead) are also of value, and
on these time scales recent observations are the most significant input to
forecasting models and more relevant than NWP. Classical time series methods
perform well (see §2.3), and those which are able to capture spatial
dependency between multiple wind farms are state-of-the-art, notably vector
autoregressive models and variants (Cavalcante et al., 2016; Messner and
Pinson, 2018). Care must be taken when implementing these models as wind power
time series are bounded by zero and the wind farm’s rated power meaning that
errors may not be assumed to be normally distributed. The use of
transformations is recommended (see also §2.2.1), though the choice of
transformation depends on the nature of individual time series (Pinson, 2012).

Wind power forecasting is reviewed in detail in Zhang et al. (2014), Giebel
and Kariniotakis (2017), Hong et al. (2020) and research is ongoing in a range
of directions including: improving accuracy and reducing uncertainty in short-
term forecasting, extending forecast horizons to weeks and months ahead, and
improving very short-term forecast with remote sensing and data sharing
(Sweeney et al., 2019, and §3.4.10).

####  3.4.7 Wave forecasting129129129This subsection was written by Jooyoung
Jeon.

Ocean waves are primarily generated by persistent winds in one direction. The
energy thus propagated by the wind is referred to as wave energy flux and
follows a linear function of wave height squared and wave period. Wave height
is typically measured as significant wave height, the average height of the
highest third of the waves. The mean wave period, typically measured in
seconds, is the average time between the arrival of consecutive crests,
whereas the peak wave period is the wave period at which the highest energy
occurs at a specific point.

The benefit of wave energy is that it requires significantly less reserve
compared to those from wind (see §3.4.6) and solar (see §3.4.8) renewable
energy sources (Hong et al., 2016). For example, the forecast error at one
hour ahead for the simulated wave farms is typically in the range of 5–7%,
while the forecast errors for solar and wind are 17 and 22% respectively
(Reikard et al., 2011). Solar power is dominated by diurnal and annual cycles
but also exhibits nonlinear variability due to factors such as cloud cover,
temperature and precipitation. Wind power is dominated by large ramp events
such as irregular transitions between states of high and low power. Wave
energy exhibits annual cycles and is generally smoother although there are
still some large transitions, particularly during the winter months. In the
first few hours of forecasting wave energy, time series models are known to be
more accurate than numerical wave prediction. Beyond these forecast horizons,
numerical wave prediction models such as SWAN (Simulating WAves Nearshore,
Booij et al., 1999) and WAVEWATCH III® (Tolman, 2008) are widely used. As
there is as yet no consensus on the most efficient model for harnessing wave
energy, potential wave energy is primarily measured with energy flux, but the
wave energy harnessed typically follows non-linear functions of wave height
and wave period in the observations of the six different types of wave energy
converters (Reikard et al., 2015).

To model the dependencies of wind speed, wave height, wave period and their
lags, Reikard et al. (2011) uses linear regressions, which were then converted
to forecasts of energy flux. Pinson et al. (2012) uses Reikard et al.’s (2011)
regression model and log-normal distribution assumptions to produce
probabilistic forecasts. López-Ruiz et al. (2016) model the temporal
dependencies of significant wave heights, peak wave periods and mean wave
direction using a vector autoregressive model, and used them to produce medium
to long term wave energy forecasts. Jeon and Taylor (2016) model the temporal
dependencies of significant wave heights and peak wave periods using a
bivariate VARMA-GARCH (see also §2.3.11) to convert the two probabilistic
forecasts into a probabilistic forecast of wave energy flux, finding this
approach worked better than either univariate modelling of wave energy flux or
bivariate modelling of wave energy flux and wind speed. Taylor and Jeon (2018)
produce probabilistic forecasts for wave heights using a bivariate VARMA-GARCH
model of wave heights and wind speeds, and using forecasts so as to optimise
decision making for scheduling offshore wind farm maintenance vessels
dispatched under stochastic uncertainty. On the same subject, Gilbert et al.
(2020b) use statistical post-processing of numerical wave predictions to
produce probabilistic forecasts of wave heights, wave periods and wave
direction and a logistic regression to determine the regime of the variables.
They further applied the Gaussian copula to model temporal dependency but this
did not improve their probabilistic forecasts of wave heights and periods.

####  3.4.8 Solar power forecasting130130130This subsection was written by
Sonia Leva.

Over the past few years, a number of forecasting techniques for photovoltaic
(PV) power systems has been developed and presented in the literature. In
general, the quantitative comparison among different forecast techniques is
challenging, as the factors influencing the performance are numerous: the
historical data, the weather forecast, the temporal horizon and resolution,
and the installation conditions. A recent review by Sobri et al. (2018)
presents a comparative analysis of previous works, also including statistical
errors. However, since the conditions and metrics used in each work were
different, the comparison is not very meaningful. Dolara et al. (2018) present
relevant evaluation metrics for PV forecasting accuracy, while Leva et al.
(2019) compare their effectiveness and immediate comprehension. In term of
forecast horizon for PV power systems, intraday (Nespoli et al., 2019) and the
24 hours of the next day (Mellit et al., 2020) are considered the most
important.

Nespoli et al. (2019) compared two of the most widely used and effective
methods for the forecasting of the PV production: a method based on Multi-
Layer Perceptron (MLP) and a hybrid method using artificial neural network
combined with clear sky solar radiation (see also §2.7.8 and §2.7.13). In the
second case, the simulations are based on a feed-forward neural network (FFNN)
but, among the inputs, the irradiation in clear sky conditions is provided.
This method is called Physical Hybrid Artificial Neural Network (PHANN) and is
graphically depicted in figure 9 (Dolara et al., 2015). PHANN method
demonstrates better performance than classical NN methods. Figure 10 shows a
comparison between the measured and forecasted hourly output power of the PV
plant for both sunny and cloudy days. The PHANN method shows good forecasting
performance, especially for sunny days.

Figure 9: Physical Hybrid Artificial Neural Network (PHANN) for PV power
forecasting. Figure 10: Measured versus forecasted output power by MLP and
PHANN methods.

Ogliari et al. (2017) compared the PV output power day-ahead forecasts
performed by deterministic (based on three and five parameters electric
equivalent circuit) and stochastic hybrid (based on artificial neural network
models) methods aiming to find the best performance conditions. In general,
there is no significant difference between the two deterministic models, with
the three-parameter approach being slightly more accurate. Figure 11 shows the
daily value of normalised mean absolute error (NMAE%) for 216 days evaluated
by using PHANN and three parameters electric circuit. The PHANN hybrid method
achieves the best forecasting results, and only a few days of training can
provide accurate forecasts.

Figure 11: Daily NMAE% of the PHANN method trained with 10 days (left) and
with 215 days (right) compared with the three-parameters model.

Dolara et al. (2018) analysed the effect of different approaches in the
composition of a training data-set for the day-ahead forecasting of PV power
production based on NN. In particular, the influence of different data-set
compositions on the forecast outcome has been investigated by increasing the
size of the training set size and by varying the lengths of the training and
validation sets, in order to assess the most effective training method of this
machine learning approach. As a general comment on the reported results, it
can be stated that a method that employs the same chronologically consecutive
samples for training is best suited when the availability of historical data
is limited (for example, in newly deployed PV plant), while training based on
randomly mixed samples method, appears to be most effective in the case of a
greater data availability. Generally speaking, ensembles composed of
independent trials are most effective.

####  3.4.9 Long-term simulation for large electrical power
systems131131131This subsection was written by Fernando Luiz Cyrino Oliveira.

In large electrical power systems with renewable energy dependence, the power
generators need to be scheduled to supply the system demand (de Queiroz,
2016). In general, for modelling long-term renewables future behaviour, such
as hydro, wind and solar photovoltaics (PV), stochastic scenarios should be
included in the scheduling, usually in a dispatch optimisation problem under
uncertainty – like described, for small systems, in §3.4.1 and, for wave
forecasting, in §3.4.7. Due to the complexity and uncertainly associated, this
problem is, in general, modelled with time series scenarios and multi-stage
stochastic approaches. de Queiroz (2016) presented a review for hydrothermal
systems, with a focus on the optimisation algorithms. §3.4.6 and §3.4.8
explore the up-to-date methods for wind and PV solar power forecasting.

Here, we emphasise the importance of forecasting with simulation in the long-
term renewable energy planning, especially in hydroelectric systems. In this
context, due to the data spatial and temporal dependence structure, time
series models are useful for future scenarios generation. Although the
proposal could be forecasting for short-term planning and scheduling (as
described in §3.4.6, 3.4.7, and §3.4.8), simulation strategies are explored
for considering and estimating uncertainty in medium and/or long-term
horizons.

According to Hipel and McLeod (1994), stochastic processes of natural
phenomena, such as the renewables ones, are, in general, stationary. One of
the main features of hydroelectric generation systems is the strong dependence
on hydrological regimes. To deal with this task, the literature focuses on two
main classes for forecasting/simulation streamflow data: physical and data-
driven models (Zhang et al., 2015). Water resources management for hydropower
generation and energy planning is one of the main challenges for decision-
makers. At large, the hydrological data are transformed into the so-called
affluent natural energy, that is used for scenarios simulation and serve as
input for the optimisation algorithms (Oliveira et al., 2015). The current
state-of-the-art models for this proposal are the periodic ones. Hipel and
McLeod (1994) presented a wide range of possibilities, but the univariate
periodic autoregressive (PAR, a periodic extension version of the ones
presented in §2.3.4) is still the benchmark, with several enhanced versions.
The approach fits a model to each period of the historical data and the
residuals are simulated to generate new future versions of the time series,
considered stationary. Among many others, important variations and alternative
proposals to PAR with bootstrap procedures (see bootstrap details in §2.7.5),
Bayesian dynamic linear models, spatial information and copulas versions (for
copulas references, see §2.4.3) are detailed in Souza et al. (2012), Marangon
Lima et al. (2014), Lohmann et al. (2016) and de Almeida Pereira and Veiga
(2019), respectively.

It is worth considering the need for renewables portfolio simulation. This led
Pinheiro Neto et al. (2017) to propose a model to integrate hydro, wind and
solar power scenarios for Brazilian data. For the Eastern United States,
Shahriari and Blumsack (2018) add to the literature on the wind, solar and
blended portfolios over several spatial and temporal scales. For China, Liu et
al. (2020) proposed a multi-variable model, with a unified framework, to
simulate wind and PV scenarios to compensate hydropower generation. However,
in light of the aforementioned, one of the key challenges and trends for
renewable electrical power systems portfolio simulation are still related to
the inclusion of exogenous variables, such as climate, meteorological,
calendar and economic ones, as mentioned in §3.4.2.

####  3.4.10 Collaborative forecasting in the energy sector132132132This
subsection was written by Ricardo Bessa.

As mentioned in §3.4.6, the combination of geographically distributed time
series data, in a collaborative forecasting (or data sharing) framework, can
deliver significant improvements in the forecasting accuracy of each
individual renewable energy power plant. The same is valid for hierarchical
load forecasting (Hong et al., 2019) and energy price forecasting (see
§3.4.2). A review of multivariate time series forecasting methods can be found
in §2.3.9 2.3.11 and §2.4.3. However, this data might have different owners,
which are unwilling to share their data due to the following reasons: (i)
personal or business sensitive information, (ii) lack of understanding about
which data can and cannot be shared, and (iii) lack of information about
economic (and technical) benefits from data sharing.

In order to tackle these limitations, recent research in energy time series
forecasting is exploring two alternative (and potentially complementary)
pathways: (i) privacy-preserving analytics, and (ii) data markets.

The role of privacy-preserving techniques applied collaborative forecasting is
to combine time series data from multiple data owners in order to improve
forecasting accuracy and keep data private at the same time. For solar energy
forecasting, Berdugo et al. (2011) described a method based on local and
global analog-search that uses solar power time series from neighbouring
sites, where only the timestamps and normalised weights (based on similarity)
are exchanged and not the time series data. Zhang and Wang (2018) proposed,
for wind energy forecasting with spatia-temporal data, a combination of ridge
linear quantile regression and Alternating Direction Method of Multipliers
(ADMM) that enables each data owner to autonomously solve its forecasting
problem, while collaborating with the others to improve forecasting accuracy.
However, as demonstrated by Gonçalves et al. (2021a), the mathematical
properties of these algorithms should be carefully analysed in order to avoid
privacy breaches (i.e., when a third party recovers the original data without
consent).

An alternative approach is to design a market (or auction) mechanism for time
series or forecasting data where the data owners are willing to sell their
private (or confidential) data in exchange for an economic compensation
(Agarwal et al., 2019). The basic concept consists in pricing data as a
function of privacy loss, but it can be also pricing data as a function of
tangible benefits such as electricity market profit maximization. Gonçalves et
al. (2021b) adapted for renewable energy forecasting the model described in
Agarwal et al. (2019), by considering the temporal nature of the data and
relating data price with the extra revenue obtained in the electricity market
due to forecasting accuracy improvement. The results showed a benefit in terms
of higher revenue resulting from the combination of electricity and data
markets. With the advent of peer-to-peer energy markets at the domestic
consumer level (Parag and Sovacool, 2016), smart meter data exchange between
peers is also expected to increase and enable collaborative forecasting
schemes. For this scenario, Yassine et al. (2015) proposed a game theory
mechanism where a energy consumer maximizes its reward by sharing consumption
data and a data aggregator can this data with a data analyst (which seeks data
with the lowest possible price).

Finally, promoting data sharing via privacy-preserving or data monetisation
can also solve data scarcity problems in some use cases of the energy sector,
such as forecasting the condition of electrical grid assets (Fan et al.,
2020). Moreover, combination of heterogeneous data sources (e.g., numerical,
textual, categorical) is a challenging and promising avenue of future research
in collaborative forecasting (Obst et al., 2019).

###  3.5 Environmental applications

####  3.5.1 Forecasting two aspects of climate change133133133This section was
written by David F. Hendry.

First into the Industrial Revolution, the UK is one of the first out: in 2013
its per capita CO2 emissions dropped below their 1860 level, despite per
capita real incomes being around 7-fold higher (Hendry, 2020). The model for
forecasting UK CO2 emissions was selected from annual data 1860-2011 on CO2
emissions, coal and oil usage, capital and GDP, their lags and non-linearities
(see §3.5.2 for higher frequency weather forecasts). Figures 12(a) to 12(c)
show the non-stationary time series with strong upward then downward trends,
punctuated by large outliers from world wars, miners strikes plus shifts from
legislation and technological change: Castle and Hendry (2020a). Saturation
estimation at 0.1% using Autometrics (Doornik, 2018) retaining all other
regressors, detected 4 step shifts coinciding with major policy interventions
like the 2008 Climate Change Act, plus numerous outliers, revealing a
cointegrated relation. The multi-step forecasts over 2012—2017 from a VAR in
panel (d) of figure 12 show the advantage of using step-indicator saturation
(SIS: Castle et al., 2015c).

Figure 12: (a) UK emissions, (b) energy sources in megatonnes (Mt) and
megatonnes of oil equivalent (Mtoe), (c) economic variables, and (d) multi-
step forecasts of CO2 emissions in Mt.

We formulated a 3-equation simultaneous model of atmospheric CO2 and Antarctic
Temperature and Ice volume over 800,000 years of Ice Ages in 1000-year
frequency (Paillard, 2001; Kaufmann and Juselius, 2013). Driven by non-linear
functions of eccentricity, obliquity, and precession (see panels (a), (b), and
(c) of figure 13 respectively), the model was selected with saturation
estimation. Earth’s orbital path is calculable into the future (Croll, 1875
and Milankovitch, 1969), allowing 100,000 years of multi-step forecasts at
endogenous emissions. Humanity has affected climate since 10 thousand years
ago (kya: Ruddiman, 2005), so we commence forecasts there. Forecasts over
$-10$ to 100 with time series from 400kya in panels (d) to (f) of figure 13
show paths within the ranges of past data $\pm$2.2SE (Pretis and Kaufmann,
2018).

Figure 13: Ice-Age data, model fits, and forecasts with endogenous CO2.

Atmospheric CO2 already exceeds 400ppm (parts per million), dramatically
outside the Ice-Age range (Sundquist and Keeling, 2009). Consequently, we
conditionally forecast the next 100,000 years, simulating the potential
climate for anthropogenic CO2 (Castle and Hendry, 2020b) noting the
‘greenhouse’ temperature is proportional to the logarithm of CO2 (Arrhenius,
1896). The orbital drivers will continue to influence all three variables but
that relation is switched off in the scenario for ‘exogenised’ CO2. The 110
dynamic forecasts conditional on 400ppm and 560ppm with $\pm$2SE bands are
shown in figure 14, panels (a) and (b) for Ice and Temperature respectively.
The resulting global temperature rises inferred from these Antarctic
temperatures would be dangerous, at more than 5∘C, with Antarctic temperatures
positive for thousands of years (Vaks et al., 2019; Pretis and Kaufmann,
2020).

Figure 14: Ice-Age simulations with exogenous CO2.

####  3.5.2 Weather forecasting134134134This subsection was written by Thordis
Thorarinsdottir.

The weather has a huge impact on our lives, affecting health, transport,
agriculture (see also §3.8.10), energy use (see also §3.4), and leisure. Since
Bjerknes (1904) introduced hydrodynamics and thermodynamics into meteorology,
weather prediction has been based on merging physical principles and
observational information. Modern weather forecasting is based on numerical
weather prediction (NWP) models that rely on accurate estimates of the current
state of the climate system, including ocean, atmosphere and land surface.
Uncertainty in these estimates is propagated through the NWP model by running
the model for an ensemble of perturbed initial states, creating a weather
forecast ensemble (Buizza, 2018; Toth and Buizza, 2019).

One principal concern in NWP modelling is that small-scale phenomena such as
clouds and convective precipitation are on too small a scale to be represented
directly in the models and must, instead, be represented by approximations
known as parameterisations. Current NWP model development aims at improving
both the grid resolution and the observational information that enters the
models (Bannister et al., 2020; Leuenberger et al., 2020). However, for fixed
computational resources, there is a trade-off between grid resolution and
ensemble size, with a larger ensemble generally providing a better estimate of
the prediction uncertainty. Recent advances furthermore include machine
learning approaches (see §2.7.10) to directly model the small-scale processes,
in particular cloud processes (see, for example, Gentine et al., 2018; Rasp et
al., 2018).

Despite rapid progress in NWP modelling, the raw ensemble forecasts exhibit
systematic errors in both magnitude and spread (Buizza, 2018). Statistical
post-processing is thus routinely used to correct systematic errors in
calibration and accuracy before a weather forecast is issued; see Vannitsem et
al. (2018) for a recent review but also §2.12.4 and §2.12.5. A fundamental
challenge here is to preserve physical consistency across space, time and
variables (see, for example, Möller et al., 2013; Schefzik et al., 2013;
Heinrich et al., 2020). This is particularly important when the weather
forecast is used as input for further prediction modelling, e.g., in
hydrometeorology (Hemri et al., 2015; Hemri, 2018).

At time scales beyond two weeks, the weather noise that arises from the growth
of the initial uncertainty, becomes large (Royer, 1993). Sources of long-range
predictability are usually associated with the existence of slowly evolving
components of the earth system, including the El Niño Southern Oscillation
(ENSO), monsoon rains, the Madden Julian Oscillation (MJO), the Indian Ocean
dipole, and the North Atlantic Oscillation (NAO), spanning a wide range of
time scales from months to decades (Vitart et al., 2012; Hoskins, 2013). It is
expected that, if a forecasting system is capable of reproducing these slowly
evolving components, they may also be able to forecast them (Van Schaeybroeck
and Vannitsem, 2018). The next step is then to find relationships between
modes of low-frequency variability and the information needed by forecast
users such as predictions of surface temperature and precipitation (Roulin and
Vannitsem, 2019; Smith et al., 2020).

####  3.5.3 Air quality forecasting135135135This subsection was written by
Claudio Carnevale.

To preserve human health, European Commission stated in the Directive
(2008/50/EC) that member states have to promptly inform the population when
the particulate matter (PM) daily mean value exceeds (or is expected to
exceed) the threshold of $50\mu g/m^{3}$. Therefore, systems have been
designed in order to produce forecasts for up to three days in advance using
as input the measured value of concentration and meteorological conditions.
These systems can be classified in (i) data-driven models (Carnevale et al.,
2016; Stadlober et al., 2018; Corani, 2005, and §2.7), and (ii) deterministic
chemical and transport models (Honoré et al., 2007; Manders et al., 2009). In
this section, a brief overview of the application of these systems to the high
polluted area of Lombardy region, in Italy, will be presented.

Carnevale et al. (2018) compared the results of three different forecasting
systems based on neural networks, lazy learning models, and regression trees
respectively. A single model has been identified for each monitoring station.
In the initial configuration, only the last three PM measurements available
were used to produce the forecast. In this configuration, the systems offered
reasonable performance, with correlation coefficients ranging from 0.6 (lazy
learning method) to 0.75 (neural network). The work also demonstrated that the
performance of the ensemble of the three systems was better than the best
model for each monitoring station (see also §2.6 for further discussions on
forecast combinations).

Starting from the results of this work, a second configuration was
implemented, using as input also the wind speed measured in the meteorological
monitoring station closest to the measurement point of PM. The researchers
observed an improvement in all performance indices, with the median of the
correlation for the best model (neural networks) increasing from 0.75 to 0.82
and the RMSE dropping from $15\mu g/m^{3}$ to $7\mu g/m^{3}$.

One of the main drawbacks of data-driven models for air quality is that they
provide information only in the point where the measurements are available. To
overcome this limitation, recent literature has presented mixed deterministic
and data-driven approaches (see, for example, Carnevale et al., 2020) which
use the data assimilation procedure and offer promising forecasting
performance.

From a practical point of view, critical issues regarding forecasting air
quality include:

  * •

Information collection and data access: even if regional authorities have to
publicly provide data and information related to air quality and meteorology,
the measured data are not usually available in real-time and the interfaces
are sometimes not automated;

  * •

Data quantity: the amount of information required by air quality forecasting
systems is usually large, in particular towards the definition of the training
and validation sets;

  * •

Non-linear relationships: the phenomenon of accumulation of pollutants in
atmosphere is usually affected by strong nonlinearities, which significantly
impact the selection of the models and their performance;

  * •

Unknown factors: it is a matter of fact that the dynamic of pollutants in
atmosphere is affected by a large number of non-measurable variables (such as
meteorological variables or the interaction with other non-measurable
pollutants), largely affecting the capability of the models to reproduce the
state of the atmosphere.

####  3.5.4 Forecasting and decision making for floods and water resources
management136136136This subsection was written by Ezio Todini.

In Water Resources and Flood Risk Management, decision makers are frequently
confronted with the need of taking the most appropriate decisions not knowing
what will occur in the future. To support their decision-making under
uncertainty, decision theory (Berger, 1985; Bernardo, 1994; DeGroot, 2004)
invokes Bayesian informed decision approaches, which find the most appropriate
decision by maximising (or minimising) the expected value of a “utility
function”, thus requiring its definition, together with the estimation of a
“predictive probability” density (Berger, 1985) due to the fact that utility
functions are rarely linear or continuous. Consequently, their expected value
does not coincide with the value assumed on the predicted “deterministic”
expected value. Accordingly, overcoming the classical 18th century
“mechanistic” view by resorting into probabilistic forecasting approaches
becomes essential (see also §2.6.2).

The failure of decision-making based on deterministic forecasts in the case of
Flood Risk Management is easily shown through a simple example. At a river
section, the future water level provided by a forecast is uncertain and can be
described by a Normal distribution with mean 10 meters and standard deviation
of 5 meters. Given a dike elevation of 10.5 meters, damages may be expected as
zero if water level falls below the dike elevation and linearly growing when