EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 1, 2017, 30-57
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Sir Clive W.J. Granger Memorial Special Issue on
Econometrics

Sir Clive W.J. Granger’s Contributions to Forecasting

Michael P. Clements

ICMA Centre, Henley Business School, University of Reading, Reading RG6 6BA, UK.

Abstract. Some of Clive Granger’s many and varied contributions to economic forecasting are
reviewed. These include contributions to forecast combination and forecast efficiency, to improving
forecast practice, to forecast evaluation, and to the theory of forecasting. We also discuss some
of the subsequent research and developments in these areas, which have sought to generalize the
applicability of Granger’s work. We also consider research in related areas motivated at least in
part by Granger’s work.

2010 Mathematics Subject Classifications: 62P20

Key Words and Phrases: Clive Granger, Forecasting, Combination, Evaluation, Forecasting
Practice

1. Introduction

Professor Clive Granger published extensively on forecasting. This paper is a personal
reflection on some of that work, and on the influence it has had on research on forecasting
and the subsequent development of the subject. The selection of topics in part reflects the
author’s own interests. It does not set out to provide a comprehensive account of Clive
Granger’s contributions to forecasting, or to catalogue his research output on this topic.
Some of these contributions were truly ground-breaking. It covers forecast combination
in section 2, improving forecast practice in section 3, forecast evaluation in section 4, and
forecasting white noise in section 5.

Email address: m.p.clements@reading.ac.uk (M. P. Clements)

http://www.ejpam.com 30 c© 2017 EJPAM All rights reserved.



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 31

2. Forecast Combination

Barnard (1963) (see [4]) pre-dates the seminal contribution of Bates and Granger (1969)
(see [5]), but the latter proposed a more general way of combining different forecasts of
the same quantity to improve predictive accuracy. [4] had considered taking a simple
average of a Box-Jenkins model forecast, and an exponential smoothing model forecast,
and showing that the resulting 1-step ahead forecasts of world airline passenger miles per
month has a smaller error variance than either set of forecasts alone. [5] generalized the
simple average to allow convex combinations of two (or more) forecasts, and to allow the
relative weights given to each to be chosen optimally based on the past performance of
the forecasts.

To illustrate, suppose there are two h-steps-ahead forecasts, f1t and f2t, of the quantity
yt. Assuming the forecasts to be unbiased, i.e. that the forecast errors eit = yt − fit
(i = 1, 2) have zero mean, Bates and Granger (1969) (see [5]) suggest the use of a combined
forecast, fct, of the form:

fct = (1− λ)f1t + λf2t. (1)

When 0 ≤ λ ≤ 1, fct comprises a simple weighted average of the two individual forecasts,
and the weighting parameter λ can be selected based on the relative accuracy of the
individual forecasts f1t and f2t.

If the forecast error associated with fct is denoted by εt = yt − fct, then the expected
squared forecast error of the combined forecast is given by:

E(ε2t ) = (1− λ)2σ21 + λ2σ22 + 2λ(1− λ)ρσ1σ2 (2)

where σ21 and σ22 denote, respectively, the expected squared errors of f1t and f2t, and ρ
denotes the correlation between the forecast errors e1t and e2t. The optimal combination
weight associated with a squared error loss function is then derived by choosing λ to
minimize (2), i.e.:

λopt = arg min
λ

{
E(ε2t )

}
=

σ21 − ρσ1σ2
σ21 + σ22 − 2ρσ1σ2

. (3)

The expected squared error associated with the optimal combination weight λopt is given
by:

E(ε2t (λopt)) =
σ21σ

2
2

(
1− ρ2

)
σ21 + σ22 − 2ρσ1σ2

where, of necessity, E(ε2t (λopt)) ≤ min
{
σ21, σ

2
2

}
. Suppose that f1 and f2 are equally

accurate, i.e., σ21 = σ22 = σ2. Then:

E(ε2t (λopt)) = σ2
(1 + ρ)

2
.

Given that |ρ| ≤ 1, then the expected squared error associated with the optimal forecast
is less than either individual forecast for all values of ρ other than ρ = 1. So there are
diversification gains in general, and in particular when the forecasts are equally accurate,



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 32

unless the forecasts are perfectly correlated. Note that when ρ = −1 a perfect forecast
results.

In practice the optimal weight parameter, and its constituent parameters ρ, σ21 and σ22,
will be estimated from the past record of forecasts and outcomes. Denoting the time series
of past h-steps-ahead forecast errors by e1t, e2t, t = 1, ..., n, the obvious sample estimator
of the population weight parameter (3) is given by:

λ̂opt =

∑n
t=1 e

2
1t −

∑n
t=1 e1te2t∑n

t=1 e
2
1t +

∑n
t=1 e

2
2t − 2

∑n
t=1 e1te2t

. (4)

This estimated weight can then be used in the future to produce out-of-sample combined
forecasts. [5] are aware that the relative performance of the component forecasts may not
be constant over time, and suggest a number of ways of allowing time dependence in λ̂
based on the past forecast errors.

An alternative but equivalent way of estimating λ is via the regression method of
Granger and Ramanathan (1984) (see [64]), that is, from ordinary least squares estimation
of:

e1t = λ(e1t − e2t) + εt (5)

or, equivalently,
yt = (1− λ)f1t + λf2t + εt. (6)

From (6) it is immediately apparent that εt is the forecast error of the forecast combination.
Hence (6) (or equivalently (5)) implies that the forecast error is uncorrelated with the
forecast combination (the explanatory variable) by construction. However, εt need not be
uncorrelated with either f1t or f2t individually, and it may be optimal to allow non-convex
combinations.

The implicit assumption behind taking convex combinations is that the forecasts are
efficient in the sense of [79]. Efficiency requires that α = 0 and β = 1 in the realization–
forecast regression yt = α+βft+εt (see, e.g., [23, Ch. 3] for a discussion), which also serves
as sufficient condition for unbiasedness. When this holds for each forecast individually,
i.e., yt = fit + εit, then (6) is warranted. But otherwise, bias can be easily accommodated
by including an intercept in (5) or (6):

e1t = α+ λ(e1t − e2t) + εt

resulting in the unbiased combination:

fct = α+ (1− λ)f1t + λf2t. (7)

More generally, forecast inefficiency suggests relaxing the assumption that the combination
weights sum to one (as advocated by [64]):

fct = α+ β1f1t + β2f2t (8)

with weights calculated from:

yt = α+ β1f1t + β2f2t + εt. (9)



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 33

Clearly (7) and (1) are special cases of (8), where the restrictions β1 + β2 = 1, and
α = 0, β1 + β2 = 1 are imposed, respectively. When the actuals and forecasts are non-
stationary integrated processes, (9) could be specified using actual and predicted changes,
rather than levels, so that the variables are I(0), and so tests will have their standard
distributions, but the outcome of the test for forecast encompassing may not be invariant
to this change. An alternative, which may be preferable when the data and forecasts are
I(1), is suggested by [44]. This amounts to regressing the forecast error for one of the two
competing forecasts on the difference between the two forecasts. Provided the forecasting
models are reasonably well specified, in the sense that both forecasts are cointegrated with
the actuals (such that the forecast errors are I(0)), then both the dependent variable and
the regressor will be I(0).

There have been a large number of extensions to the analysis discussed here, including:
allowing for autocorrelation in εt when estimating combination weights (e.g., [33] and
[27]); allowing weights to vary over time (e.g., [34], [75], and [32]); simple averaging versus
estimating weights (e.g., [76], [91], [46] and [48]); Bayesian combination methods (e.g., [13],
[35] and [78]); combination for interval, density and probability forecasts (e.g., [99], [80]
and [20]) and nonlinear combinations (e.g., via Artificial Neural Networks, [37]). Many
Handbooks provide chapters reviewing this literature, including [85], [96], [19] and [1].

The notion of conditional efficiency was developed by Nelson (1972) (see [83]) and
Granger and Newbold (1973) (see [58]) to denote forecasts which could not be made more
accurate by combination with another forecast. More precisely, a forecast f1 is said to
be conditionally efficient with respect to f2 if the optimal weight on f2 in a combination
with f1 is zero. [11] interpreted conditional efficiency in terms of the wider concept of
encompassing (see, inter alia, [81], [82] and [72]) as forecast encompassing (see, e.g., [45],
[2] and [70] for further developments and applications).

Finally, Granger (1989) (see [51] considers the relationship between pooling forecasts
and pooling information. The Bates and Granger (1969) (see [5]) example considered below
of combining linear and exponential trend models of an output index assumes both models
have access to the same information set (just the variable in question, the output index),
and the potential for combination arises through the use of different model specifications.
Alternatively, the more generic case is perhaps where different models (or forecasters)
have access to different information sets. Granger supposes each forecaster j = 1, . . . , J
has access to a specific information set, Ij,t, and all share the common information, Io,t.
We assume each information set consists of a single variable, and its lags, and all these
variables are uncorrelated with each other (at all leads and lags), and with the common
variable known to all: Ij,t = {xj,t, xj,t−1, . . .}, Io,t = (xt, xt−1, . . .), and Cov (xj,t, xi,s) = 0
for all t and s when i 6= j, and Cov (xt, xi,s) = 0 for all i, s and t.

Given all the information, Ωt = {I1,t, I2,t, . . . , IJ,t, Io,t}, suppose the optimal 1-step

ahead forecast takes the form E (yt+1|Ωt) = α (L)xt +
∑J

j=1 βj (L)xj,t. Each forecaster
will report their conditional expectation, given by: E (yt+1|Ij,t) = α (L)xt + βj (L)xj,t.



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 34

Faced with the J-forecasts, the equally-weighted forecast combination is given by:

fc,t+1 = α (L)xt +
1

J

J∑
j=1

βj (L)xj,t.

so that the individual xj,t are weighted by J−1 (as opposed to 1 in the ‘pooling of infor-
mation’ case), and for large J , fc,t+1 will approach fo,t+1 = E (yt+1|Io,t) = α (L)xt. If
fo,t+1 were available, then the combination Jfc,t+1− (J − 1) fo,t+1 matches the pooling of
information.

This shows that aggregating information, E (yt+1|Ωt), is not equivalent to aggregating
forecasts, fc,t+1. This holds in general and not just for equally-weighted forecasts. The
problem is that the forecast combination does not use the information efficiently. In the
simple example considered here, a second-level of combination with the new forecast fo,t+1

results in efficient combination relative to the pooling of information.
There is now a large literature on forecasting when there are many potential predictors.

[92] is a good example of this literature, showing that combinations of individually unstable
forecasts offer some improvement in forecast accuracy over univariate benchmarks. [18]
directly compared combining forecasts and combining information in modelling. Factor
models are a way of using the information from many predictor variables in a forecasting
model (see, e.g., [47] and [93]), and [31] for an alternative approach.

We conclude the discussion of forecast combination with i) a re-analysis of an empirical
example of forecast combination in [5], based on [71], followed by ii) a brief review of recent
work allowing for more general loss functions, that is, going beyond the squared-error loss
function used by Bates and Granger (1969) (see [5]) and Granger and Ramanathan (1984)
(see [64]).

2.1. Forecast Combination Example

[71] re-visit the forecast combination example of [5, Table A1, p. 462], which considers
linear and exponential trend models of an output index for the gas, electricity and water
sector. Table 1 records the output index for the years 1948 to 1965, along with forecast
errors from linear and exponential trend models of output {yt}, given by:

yt = α+ βt+ errort,

ln (yt) = a+ bt+ errort

where t is a linear time trend. The forecast errors in each period t (t = 1950, . . . , 1965) are
for forecasts based only on models estimated up to t− 1. The exponential model forecasts
are clearly superior on the sum of squared errors (SSE), and therefore on commonly-used
forecast accuracy measures such as the (root) mean squared forecast error. Nevertheless,
a combination of the exponential and linear trend models has a smaller SSE. A weight of



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 35

0.16 on the linear forecasts results in a combined forecast SSE of 78.8, compared to an
SSE for the exponential trend model of 84.4. ∗

[71] consider forecast combination from the perspective of the theory of forecasting
enunciated in [23] and [24] (and subsequently refined in, e.g., [10]). This theory replaces the
traditional assumptions of i) the model being essentially complete and correctly-specified
for the variables of interest and ii) that model remaining constant over the period being
forecast, with their obverses: a) models are incomplete and incorrect in many ways, and
b) the processes being modelled will typically exhibit evolution over time as well as abrupt
shifts and changes. Their paper suggests a rationale for forecast combination in terms of
the component models being differentially susceptible to structural breaks.

In terms of the output index illustration, it is evident that the ‘constant absolute
increase’ implication of the linear trend model is inappropriate - the forecast errors become
large and positive from around 1961 onwards. On average, the exponential model generates
negative errors, and combination is seen to work by averaging the over-predictions of the
more-accurate exponential model with the under-predictions of the linear trend model
over the 1955 – 61 period. Suppose the component forecasts are first bias corrected. The
results of bias-correcting the individual forecasts and their SSEs are shown in the last two
columns of the table. The bias-correction is calculated in real time, in the sense that the
forecast of period t is calculated by adding the sample mean of the forecast errors up to
period t − 1. This is a feasible correction, in that it uses only information available at
each forecast origin, but it results in slow adaptation of the forecasts to past systematic
errors. Nevertheless, the SSE of the bias-corrected exponential forecasts is 77, less than
the combined forecast SSE of 78.8 (with a weight of 0.16). Any fixed-weight combination
of the bias-corrected forecasts, with weights in the interval (0, 1), has a larger SSE than
that of the exponential model forecasts. The optimal weight on the linear forecasts (after
bias-correcting, and imposing the constraint that they sum to unity) was −0.22, with an
SSE of 72.61. Negative weights may appear anomalous, but see [96]. The fixed-weight
combination forecasts are not feasible, as they are calculated based on the full set of
forecast errors, and as noted above, [5] suggest time-varying weight schemes.

The [71] extension to the forecast combination example serves to illustrate that gains
from combination may result from models with manifestly mis-specified deterministic com-
ponents. This is consistent with the primacy of breaks or shifts in deterministic factors
in causing forecast failure, and the closely-related effects of the mis-specification of such
components (see, e.g., the forecast-error taxonomy in [26]). We consider this example
again in section 3 in the context of intercept correction.

2.2. Loss Functions and Forecast Combination

The early work on forecast combination of [5] and [64] assumed combination weights
chosen to minimize a symmetric, squared-error loss function, and out-of-sample combi-

∗The numbers are the calculations of [71], who calculate the forecasts and statistics reported in the table
from the actual series. Small differences relative to Bates and Granger’s figures were attributed to improved
precision.



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 36

Table 1: Forecast errors of output indices, 1950–65

Actual 1-step forecast errors
Linear Exponential Combination Linear Exponential

Bias-corrected Bias-corrected

1948 58.0
1949 62.0
1950 67.0 1.0 0.7 0.77 1.0 0.7
1951 72.0 0.7 0.1 0.21 -0.3 -0.6
1952 74.0 -2.5 -3.4 -3.24 -3.3 -3.8
1953 77.0 -2.2 -3.3 -3.11 -1.9 -2.4
1954 84.0 2.1 0.8 0.99 2.8 2.2
1955 88.0 1.0 -0.6 -0.37 1.2 0.4
1956 92.0 0.4 -1.7 -1.33 0.4 -0.7
1957 96.0 0.0 -2.5 -2.08 -0.0 -1.4
1958 100.0 -0.2 -3.2 -2.71 -0.3 -2.0
1959 103.0 -1.3 -4.8 -4.28 -1.4 -3.4
1960 110.0 1.9 -2.1 -1.47 2.0 -0.3
1961 116.0 3.2 -1.4 -0.71 3.1 0.4
1962 125.0 7.0 1.8 2.60 6.7 3.5
1963 133.0 8.8 2.8 3.74 8.0 4.3
1964 137.0 6.1 -0.9 0.26 4.7 0.3
1965 145.0 8.0 -0.0 1.26 6.3 1.1

Sample bias 2.1 -1.1 -0.6 1.8 -0.1
Sum of squared errors 263.3 84.4 78.8 211.9 77.0

The output series is the output index for the gas, electricity and water sector, given in [5,
Table A1, p. 462]. The combination forecast has fixed weights of 0.16 and 0.84 on the
(uncorrected) linear and exponential forecasts

nations of forecasts would typically be assessed in terms of the (R)MSFE, the empirical
counterpart of expected squared-error loss. The earlier analysis has subsequently been
generalized to allow for asymmetric loss by [41]. For general loss functions and forecast
error distributions, the optimal combination weights can be shown to depend on higher-
order moments of the forecast error distribution, although under certain restrictions on the
form of the forecast error distribution, the optimal combination weights on the individual
forecasts are identical to the squared-error loss weights for almost all loss functions - only
the value of the constant term in the combination will differ. Beginning with the joint
distribution of the actual and forecasts (yt, ft)

′, [41] show that the restrictions require that
the marginal distribution of the forecast errors depends only on the first two moments
of the forecast errors. This holds when the joint distribution is elliptically symmetric
(which includes the multivariate normal and t-distributions). Under these assumptions,
the squared-error loss weights of [5] remain relevant, and only the constant term differs.

To understand this result, we sketch the analysis in [41]. The notation for the first two



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 37

moments of the actuals and forecasts are given by:

E

(
yt
ft

)
=

(
µy
µ

)
, Cov

(
yt
ft

)
=

(
σ2y σ′21
σ21 Σ22

)
.

Write the forecast combination error as et = yt−β0−β′ft to define the combination weight
vector as β, and the intercept in the combination by β0. Then the mean and variance of
the combination forecast error are given by:

µe = µy − β0 − β′µ (10)

σ2e = σ2y + β′Σ22β−2β′σ21. (11)

Suppose the loss function is L (et), where L (et) = e2t gives the standard squared-error loss.
The forecast combination is defined by (β0, β), and the optimal combination minimizes
the expected loss, E [L (et)], i.e.,:

min
β0,β

∫
L (et) dF (et) .

Under elliptical symmetry, we can write E [L (et)] = g
(
µe, σ

2
e

)
. From (10) and (11), only

µe depends on β0. Thus the first order condition for minimizing E [L (et)] with respect to
β0 is:

∂g
(
µe, σ

2
e

)
∂β0

=
∂g
(
µe, σ

2
e

)
∂µe

∂µe
∂β0

= 0.

As ∂µe
∂β0

= −1, the optimal value for β0, β
∗
0 , solves

∂g(µe,σ2
e)

∂µe
= 0. β∗0 depends on L (.), and is

set to generate the optimal amount of bias (µ∗e) given the form of L (.). For squared-error

loss, E [L (et)] = µ2e + σ2e , and
∂g(µe,σ2

e)
∂β0

= −2µe, so that the optimal amount of bias is of
course zero (µ∗e = 0).

Consider the first order condition with respect to β:

∂g
(
µe, σ

2
e

)
∂β

=
∂g
(
µe, σ

2
e

)
∂σ2e

∂σ2e
∂β

= 0.

Provided
∂g(µe,σ2

e)
∂σ2

e
6= 0,

∂g(µe,σ2
e)

∂β = 0 implies that ∂σ2
e

∂β = 0, so from (11), 2Σ22β
∗= 2σ21,

and β∗ = Σ−122 σ21 irrespective of the form of L (.), matching the expression for squared-
error loss.

As [41] remark, if an element of β∗ is zero under squared-error loss, then the corre-
sponding forecast will also receive zero weight under any other loss function, assuming
that the stated properties of the forecast error distribution hold.



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 38

3. Improving Forecasting Practice

Granger wrote or co-authored a number of papers on forecasting practice, indicating
how the then current approaches might be improved, both from a technical perspective,
and in terms of how forecasts are presented. A number of recommendations were made
which helped foster marked improvements in the practice of forecasting, and forecasting
research.† For example, Granger and Newbold (1973) (see [58]) are critical of statistical
forecast evaluation criteria which are not monotonic functions of squared-error loss, such
as the ‘inequality coefficient’ (or first U -statistic) of [94]. They also criticise the practice of
comparing the distributional and time-series properties of the actual and forecast series,
and suggest instead a consideration of the forecast error, and especially the one-step ahead
error, as this has readily-testable properties under forecast optimality (e.g., unbiasedness,
serially uncorrelatedness: but see section 4.2). When feasible, forecasts should be eval-
uated in terms of the ‘expected utility’ resulting from actions or decisions taken on the
basis of those forecasts, as in Granger and Pesaran (2000,2000) (see [63, 62]) (discussed in
section 4.3).

Granger and Newbold (1973,1975) (see [58, 60]) suggested econometricians should pay
more attention to time-series models, arguing that ‘Econometricians rarely, if ever, con-
sider the problem of forecasting a time series in terms of its current and past values....if
they did so they might learn a number of valuable lessons applicable to the more so-
phisticated model building exercises they attempt’ ([58]). It is not sufficient to simply
show that ‘econometric forecasts’ outperform ‘extrapolative forecasts’ (i.e. Box-Jenkin
forecasts), but the econometric forecasts should be conditionally efficient (as defined in
section 2) with respect to multivariate Box-Jenkins forecasts. There is a call for more
stringent comparators than ‘no change’ or ‘same change’ predictors, and an emphasis on
the forecasts embodying all the useful information in the ‘purely statistical’ forecasting
devices. It is argued that econometric model forecasts would typically contain extraneous,
non-numerical information which could not be easily accommodated in the Box-Jenkins
models, so that such models would necessarily be at an advantage, and simply being more
accurate would not be sufficiently demanding.

In addition, Granger and Newbold (1974) (see [59]) advocated the use of differenced-
data in dynamic modelling as a way of countering the finding of nonsense or spurious
regressions: see the paper by Jennifer Castle and David Hendry in this volume. The
forecasting implications of the treatment of unit roots was subsequently discussed by [43],
[21] and [38], inter alia.

The call for greater emphasis on the dynamic relationships between variables would
appear to have been answered in [90], and the subsequent popularity of vector autoregres-
sions in macro-econometric modelling and forecasting (see, e.g., [36]).

Some 20 years later, Granger (1996) (see [53]) argued that it was usually the case that
i) forecasts were published with no indication of the level of uncertainty, ii) there was
no recognition of the differing degrees of difficulty in forecasting different variables, iii)
or of whether the forecasts were conditional or unconditional, iv) or of the role of data

†We consider one of his recommendations separately in section 5



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 39

revisions and seasonal adjustment, v) or of the techniques used, the assumptions made,
or the information set, vi) that there was a tendency of forecasters to ‘all swing together’,
and vii) and a tendency to under-estimate change.

Since then, there has been substantial progress on many of these. For example, there
has been much work on the calculation and representation of forecast uncertainty (i),
both in academia and amongst policy makers (see, e.g., [89], [74] and [69]), and in terms
of the effects of data revisions (iv), facilitated by the availability of ready-made real-time
datasets ([30]): see, for example, the review chapters by [28, 29]. There has also been
research on the intersection of these two - looking at the impact of data revisions on
forecast uncertainty: [17]. There has also been considerable progress on other entries in
Granger’s list.

Granger applauded the use of past forecast errors as a method of intercept-correction,
or putting the forecast ‘back-on-track’, as advocated in [22] and discussed more fully in
[24, ch. 6], as a way of improving forecasting practice.

The role of intercept correction in mitigating the effects of structural breaks and dy-
namic mis-specification are explained by [24] as follows. Suppose the forecaster assumes
yt is generated by

yt = µ+ ut where ut ∼ N
(
0, σ2u

)
(12)

and µ is estimated from a sample of size T by least squares, and used to forecast:

yT+1|T = µ̂ = T−1
T∑
t=1

yt. (13)

When (12) is the actual process that generates the data, yT+1|T is the estimated conditional
expectation, E (yT+1 | yT ) = µ. But suppose there is error autocorrelation, as in the linear
trend model of section 2, and in addition a shift in E (yt) at time T1, since the actual data
generating mechanism is:

yt = µ+ δ1{t≥T1} + ρyt−1 + εt where εt ∼ IN
(
0, σ2ε

)
, (14)

where |ρ| < 1, and 1{t≥T1} = 0 for t < T1, and otherwise 1{t≥T1} = 1. Hence E (yT+1) =
(µ+ δ) / (1− ρ) and E (yT+1 | yT ) = µ+δ+ρyT , but the mis-specified model has a forecast
mean of:

E (µ̂) = T−1

 T1∑
t=1

E (yt) +
T∑

t=T1+1

E (yt)

 ≈ µ+ δ

1− ρ
− κ δ

1− ρ
, (15)

where κ = T−1T1, and T1 ≤ T . When T1 = T , κ = 1, and E (µ̂) = µ (1− ρ)−1, and
the in-sample estimate reflects none of the changed intercept. (The approximation sign in
(15) reflects the fact that ‘E (yt)’ is not constant for t ≥ T1, but adjusts over time, given
(14), but we suppress this in what follows in the interests of simplicitly).

Setting the model ‘back on track’ adds to the forecast the forecast-origin error, ûT =
yT − µ̂ (and so fits the last observation perfectly), yielding:



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 40

ŷι,T+1 = µ̂+ ûT = yT . (16)

Note that:

E (ŷι,T+1) = E (yT ) =
µ+ δ

1− ρ
,

which is unconditionally unbiased, despite the model being mis-specified. Further:

yT+1 − ŷι,T+1 = µ+ δ + (ρ− 1) yT + εT+1

= (ρ− 1)

(
yT −

µ+ δ

1− ρ

)
+ εT+1,

so that the unconditional expected squared-error is:

E
[
(yT+1 − ŷι,T+1)

2
]

= σ2ε + (1− ρ)2 V ar (yT ) =
2σ2ε

1 + ρ
, (17)

as against the minimum obtainable (under correct specification, a known break, and known
parameters) of σ2ε .

Intercept correction in this example yields unbiased forecasts, with an increase in the
forecast-error variance which is decreasing in ρ, provided ρ > 0, that is, the variance cost
is decreasing in the degree of dynamic mis-specification. In general, for models which
are not dynamically-mis-specified, the overall efficacy of intercept correction in terms of
expected squared error will rely on a favourable tradeoff between an inflated forecast-error
variance and reduced (squared) bias.

In terms of the output index illustration recorded in table 1, it is apparent that the
linear trend model generates a sequence of positive forecast errors over the second half of
the period (even after bias-correcting), which is consistent with an inappropriate specifi-
cation of the deterministic term andor a shift in the trend function. Either way, the above
analysis suggests the use of an intercept correction. Adding in the last error reduces the
linear model (bias-corrected) sum of squares from the value of 211.9 shown in the table to
121.1. This strategy is not successful for the non-linear trend model - the sum of squares
is increased from 77.0 to 111.4. The forecasts from this model do not exhibit systematic
bias, and there is therefore no bias offset to the higher forecast-error variance.

4. Forecast Evaluation

4.1. Generalized Cost of Error

Granger (1969) (see [49]) was a key paper in the development of a prediction the-
ory which generalizes the quadratic forecast loss functions of classical prediction theory.
Granger argued that in practice - at least in the fields of economics and management -
the ‘cost of error’ may not be proportional to the squared forecast error, and cited two



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 41

motivating examples.‡) Granger comes up with the practical recommendation that when
the loss function is symmetric (but not quadratic), one may use the least-squares predictor
(that is, proceed as one would under squared-error loss), and that when the loss function
is non-symmetric, a simple constant bias term added to the predictor will be a reasonable
solution. More precisely, assuming Gaussianity, and a symmetric loss function, the least
squares predictor will not be optimal but will constitute an ‘efficient and computationally
simple method’. If the loss function is non-symmetric, the least squares predictor can be
adjusted by a constant amount at a second stage. The constant is chosen to minimize the
sum of the losses over a sample of actual values and (least-squares) forecasts. Granger
suggests these prescriptions may be sensible even when the data are non-Gaussian. When
the data are non-Gaussian, the optimal predictor will not necessarily be a linear function
of the past data, and so the use of the linear least squares predictor will be non-optimal.
Building on the earlier work of [101], Granger argues the use of linear predictors may
yield reasonable approximations, particularly if there is no information on the form of the
non-linearity (which we interpret as absent knowledge of the relevant non-linear model of
the sort surveyed in [65], and discussed in the paper by Timo Teräsvirta in this special
issue).

Granger (1969) (see [49]) details the derivation of the optimal predictor for a non-
symmetric linear loss function. Firstly, in terms of the notation of that paper, the optimal
predictor h, where h is some function of the data Xt, Xt−1, . . ., is given by:

E [g (Xt+k − h) | Xt, Xt−1, . . .] = Ec [(g (Xt+k − h))] (18)

=

∫ ∞
−∞

g (x− h) fc (x) dx (19)

where g () is the loss function, and fc is the conditional density function of Xt+k. For
squared-error loss, g (x) = x2, and writing (Xt+k − h) = (Xt+k −M) + (M − h), where
M = Ec (Xt+k) is the conditional expectation:

Ec

[
(Xt+k − h)2

]
= Ec

[
(Xt+k −M)2

]
+ Ec

[
(M − h)2

]
+ 2Ec [(Xt+k −M) (M − h)]

= Ec

[
(Xt+k −M)2

]
+ Ec

[
(M − h)2

]
(the last term in the first line is necessarily zero - the conditional-expectation forecast error
is uncorrelated with forecast origin functions M and h). The expression is minimized by
h = M , showing that the conditional expectation is the optimal predictor for squared-error

‡In the first a bank decides how large a computer to buy to handle its current accounts, with an over-
prediction of the bank’s future business resulting in too large and expensive a computer, and an under-
prediction a computer unable to handle all the accounts, and Granger suggests there is no reason to expect
under- and over-predictions of the same magnitude would be equally costly. The second example concerns
a bank acting as an issuing agent for a share issue, where too high a price leaves the bank liable to purchase
the shares at a high price and too low a price will reduce the amount of money made by the issuing firm.
Again, there is no reason to think the costs are symmetric. These examples suggest in real-world situations
costs are unlikely to be symmetric (or, presumably, necessarily quadratic), but Granger was also aware
that ‘real-world cost functions are rarely available’. Granger (1993) (see [52, p.651])



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 42

loss. When Xt is Gaussian, M is a linear function of Xt, Xt−1, . . ., and fc (x) is normal
with mean M .

When loss is non-symmetric but linear, e.g.,

g (x) = ax, x > 0, a > 0
= 0, x = 0
= bx, x < 0 b < 0

from (19) Granger obtains:

Ec [(g (Xt+k − h))] = a

∫ ∞
h

(x− h) fc (x) dx+ b

∫ h

−∞
(x− h) fc (x) dx.

Setting the derivative with respect to h to zero implicitly defines h via Fc (h) = a (a− b)−1,
where Fc (x) is the c.d.f. of Xt+k. Assuming Gaussianity, h = M + α where α does not
depend on Xt, Xt−1, . . . .. Granger’s suggestion is that in practice it may generally be
reasonable to assume Gaussianity holds, in which case the optimal predictor is a simple
constant adjustment to the conditional expectation, as here.

Granger’s ideas have been extended to allow for processes which are conditionally-
Gaussian, which allow for time-varying forecast-error variances instead of a constant con-
ditional variance, allowing for ARCH and GARCH processes (e.g., [42] and [6]). [12] show
that allowing for time-varying forecast-error variances gives rise to an adjustment which is
no longer constant, but instead depends on the forecast variance of the process. Consider
the ‘linex’ loss function of [98], which is a popular choice in the literature as it permits a
closed-form solution for the optimal predictor:

C
(
et+k|t

)
= b

[
exp

(
aet+k|t

)
− aet+k|t − 1

]
, a 6= 0, b ≥ 0

where et+k|t is the forecast error, and the notation makes explicit the target period and fore-
cast origin as t+k and t, respectively. For a > 0, the loss function is approximately linear
for et+k|t < 0 (‘over-predictions’), and exponential for et+k|t > 0, (‘under-predictions’),
and conversely for a < 0. Assume the process being forecast is conditionally Gaussian:

yt+k | It ∼ N
(
yt+k|t, σ

2
t+k|t

)
,

then the optimal predictor ŷt+k|t can be shown to be:

ŷt+k|t = yt+k|t +
a

2
σ2t+k|t. (20)

where yt+k|t = E (yt+k | It) is the conditional expectation, and the ‘adjustment term’
a
2σ

2
t+k|t depends on the the degree of asymmetry a, the forecast horizon k, and the past

data (the last two through the forecast of the variance, σ2t+k|t). In Granger’s setup the

assumption that σ2t+k|t = σ2k gives rise to a constant adjustment (given a and k). For small

a, the loss function is approximately quadratic (from a Taylor-series expansion of C (e)

about e = 0) , C (e) ' ba2

2 e
2, and so ŷt+k|t → yt+k|t as the degree of asymmetry lessens.



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 43

Recently [87] have shown that essentially the same results hold for general asymmetric
loss functions, under relatively weak conditions on the form of the loss function and the
data generating process. The condition on the data generating process is that the variable
of interest is conditionally location-scale distributed, whereas the loss function must be
homogeneous in the forecast error.

Formally, for some constant distribution function D, we need to assume:

yt+k | It ∼ D
(
yt+k|t, σ

2
t+k|t

)
,

where, as above, yt+k|t = E (yt+k | It) and σ2t+k|t = V ar (yt+k | It) and the loss function
satisfies:

L
(
a · et+k|t

)
= g (a)L

(
et+k|t

)
,

for some positive function g, and all a 6= 0. [87, Proposition 2] show that the optimal
forecast is given by:

ŷt+k|t = yt+k|t + φhσt+k|t (21)

where φh is a constant that depends on the form of D and L.§

From (21) it follows immediately that:

E
(
yt+k − ŷt+k|t | It

)
= E

[
yt+k −

(
yt+k|t + φhσt+k|t

)
| It
]

= −φhσt+k|t
so that optimal forecasts are (conditionally) biased. But although the bias of a rational
forecaster should depend on the forecast standard deviation, it should not depend on other
variables known at time t. This suggests testing for rational expectations with asymmetric
losses by running a regression such as:

et+k|t ≡ yt+k − ŷt+k|t = ζ1σt+k|t + ζ ′2Zt + εt+k (22)

where Zt is a vector of variables known at time t, Zt ⊂ It. Under the null of efficient use of
information - in the sense that forecasts cannot be systematically improved using forecast-
origin information - we would expect to find ζ2 = 0, but ζ1 6= 0 if loss is asymmetric: see
e.g., [88].

Asymmetric loss has been widely used in a number of contexts: to consider whether
apparently biased and inefficient forecasts are consistent with rational behaviour (see, e.g.,
[39, 40]); as an explanation of the observed dispersion of inflation expectations (resulting
from heterogeneity in the degree of asymmetry of individuals’ loss functions) ([8]); and
as a possible explanation of apparent inconsistencies between survey respondents’ point
predictions and probability distributions (see, e.g., [15, 16]); amongst many others.

[73] stress that the calculation of the optimal predictor requires knowledge of the con-
ditional distribution over which the integration is assumed to be carried out. For example,

§Homogeneity of the loss function rules out linex loss. A practical implication of adopting the [87] frame-
work rather than linex loss (together with the assumption that the data generating process is conditionally
normal, in order to obtain an expression for the optimal predictor) is that the optimal predictor should
depend linearly on the conditional standard deviation, rather than the conditional variance.



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 44

the calculation of yt+k|t = E (yt+k | It) requires knowledge of the conditional distribution
of yt+k given It. This requires stationarity, so that the distribution can be calculated
from past data. In the event of shifts in the underlying distributions, expectations based
on now outdated distributions will not be ‘optimal’, and may create a role for the robust
forecasting devices recently reviewed by [9] and [10].

4.2. Properties of Optimal Forecasts

Granger (1999) (see [54]) recognized that the standard properties of optimal forecasts
only applied for a squared-error loss function, and outlined a forecast theory for general-
ized cost functions. Under squared-error loss, it is straight forward to show that optimal
forecasts are unbiased, that the forecast-error variance is monotonically non-decreasing in
the forecast horizon (at least in population), and that k-step ahead forecast errors can be
written as a moving-average process which is at most of order k − 1 (and thus one-step
forecast errors are serially-uncorrelated, as alluded to in section 3). These theoretical
properties can be used to test the optimality of empirical forecasts. However, under more
general loss functions than squared-error loss, these properties may no longer apply. As an
example, the section 4.1 shows that optimal forecasts will typically be biased under asym-
metric loss: equation (20) shows that the optimal predictor differs from the conditional
expectation - the unbiased optimal predictor under squared-error loss.

Given a general cost function C (e) defined on the forecast error e, Granger suggests
taking the derivative of the loss function with respect to the forecast, evaluated at the
forecast error corresponding to the optimal forecast. It is straightforward to derive some
of the properties of this ‘generalized forecast error’ from the first-order condition which
defines the optimal predictor. The optimal predictor is defined by:

arg min
ŷt+1|t

C
(
yt+1 − ŷt+1|t

)
dPt+1|t (yt+1)

where Pt+1|t is the conditional c.d.f. The first-order condition is:

C ′dPt+1|t (yt+1)

Granger sets Zt+1|t = C ′
(
yt+1 − ŷt+1|t

)
, and it follows immediately that Zt+1|t is condi-

tionally unbiased E
(
Zt+1|t | It

)
= 0, and therefore unconditionally unbiased, E

(
Zt+1|t

)
= 0. Moreover, E

(
Zt+1|t | It

)
= 0 implies that:

E
(
Zt+1|tWt | It

)
= 0

where Wt ⊂ It (and more generally E
(
Zt+1|tg(Wt) | It

)
= 0) for any finite function g (.)),

which suggests that in the test regression:

Zt+1|t = α+ βWt + εt+1

under optimality both α = 0 and β = 0. A valid choice for (an element of) Wt would be
the explanatory variable σt+k|t used in the test regression (22), for example, given that



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 45

the forecast standard deviation is based on information in It. Hence the switch from
the forecast error to the derivative of the cost function allows simple tests of whether
the forecast efficiently uses forecast-origin information. Since Zt|t−1 ⊂ It, and et|t−1 =
yt − ŷt|t−1 ⊂ It, Zt+1|t should be uncorrelated with both Zt|t−1 and et|t−1.

Granger shows that the results generalize beyond the one-step horizon, so that:

E
(
Zt+k|tg(Wt) | It

)
= 0

showing unbiasedness (by setting g(Wt) = 1) and forecast efficiency (by setting g(Wt) =
ŷt+k|t). In addition, Zt+k|t will be autocorrelated at most up to k − 1, matching the
standard result for the forecast error from an optimal forecast under squared-error loss.

[86] provide some extensions, and [3] an empirical application.

4.3. A Decision-theoretic Approach to Forecast Evaluation

Granger and Pesaran (2000,2000) (see [63, 62]) and Granger and Machina (2006) (see
[56]) consider evaluating forecasts in terms of the expected economic value emanating from
actions taken based on those forecasts. The paper by [56] reviews earlier contributions,
dating back to [95], and including Bayesian decision analysis (see, e.g., [100]).

[62] review the classic case of a simple two state - two action model, which establishes
the key ideas. It establishes the advantages of using a decision-theoretic approach. The
two possible states in period t+ 1 are “Bad” (st+1 = 1) and “Good” (st+1 = 0), and there
are two possible actions open to the decision-maker in period t. To take action, “Yes”,
indicated by yt = 1, or to decline to take action, “No”, yt = 0. Thus, actions are taken in
advance. The payoff matrix associated with this decision problem is given in table 2. In
the Payoff Matrix, Ut+1,by represents the decision maker’s utility if the bad state occurs
after the yes decision is taken, and so on.

Table 2: Payoff Matrix for a Two-State, Two-Action Decision Problem

States ( st+1)
Bad (st+1 = 1) Good (st+1 = 0)

Decisions (yt) Yes (yt = 1) Ut+1,by Ut+1,gy

No (yt = 0) Ut+1,bn Ut+1,gn

Table 2 reports the payoffs (U) from combining outcomes (or realized states) and
(prior) actions to economic values. A forecast probability at period t of the Bad event
occurring in period t + 1 can be used by the decision maker of a particular forecast
probability to determine their action. Let πt+1 be the actual probability that st+1 = 1,
πt+1 = Pr (st+1 = 1), and π̂t+1 the forecast probability. Probabilities of states are assumed
independent of actions. Then, the expected utility of taking action (yt = 1) based on the
forecast probabilities is given by:

Ut+1,byπ̂t+1 + Ut+1,gy (1− π̂t+1) (23)



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 46

and of not acting:
Ut+1,bnπ̂t+1 + Ut+1,gn (1− π̂t+1) . (24)

The decision maker will taken the action (yt = 1) if (23) exceeds (24), that is, if:

π̂t+1 > qt+1

where:

qt+1 =
δt+1,g

δt+1,g + δt+1,b
, and δt+1,g = Ut+1,gn − Ut+1,gy, δt+1,b = Ut+1,by − Ut+1,bn.

It is assumed that qt+1 > 0, because δt+1,b > 0 and δt+1,g > 0 by assumption. Uby > Ubn
because ‘action’ will alleviate the costs incurred in the bad state (having gritted the roads
when there is a frost will reduce the number of road traffic accidents). Ugn > Ugy because
gritting is costly and unnecessary when there is not a frost.

Hence the decision rule is y∗t = 1(π̂t+1 > qt+1). The economic benefit that accrues at
period t+ 1 will depend on which state materialises and the action taken at period t:

υt+1 (yt; st+1) = Ut+1,byst+1yt + Ut+1,gy (1− st+1) yt

+Ut+1,bnst+1 (1− yt) + Ut+1,gn (1− st+1) (1− yt)

Using the optimal decision rule:

υt+1 (y∗t ; st+1) = Ut+1,byst+1y
∗
t + Ut+1,gy (1− st+1) y

∗
t

+Ut+1,bnst+1 (1− y∗t ) + Ut+1,gn (1− st+1) (1− y∗t ) . (25)

If we substitute y∗t = 1(π̂t+1 > qt+1), we obtain υt+1 as a function of the forecast proba-
bility, υt+1 (π̂t+1; st+1):

υt+1(π̂t+1; st+1) = at+1 + bt+1(st+1 − qt+1)1 (π̂t+1 > qt+1) , (26)

where at+1 = st+1Ut+1,bn+(1−st+1)Ut+1,gn and bt+1 = Ut+1,by−Ut+1,bn+Ut+1,gn−Ut+1,gy.
Since the part of the economic value given by at+1 does not depend on the probability
forecast estimate, π̂t+1, it can be ignored when comparing two or more rival forecast
probabilities (say, π̂t+1 and π̃t+1).

The expected economic value of using the probability forecast π̂t+1 is given by

E [υt+1(π̂t+1; st+1) | Ωt] = E (at+1 | Ωt) + bt+1 (πt+1 − qt+1) 1(π̂t+1 > qt+1) (27)

where expectations are taken with respect to the true conditional probability distribution
of st+1, denoted by E (· | Ωt), and πt+1 = E (st+1 | Ωt) = Pr (st+1 = 1 | Ωt), 1 − πt+1 =
Pr (st+1 = 0 | Ωt).

Ignoring the dependence of probabilities on actions, from (27) the part of the expected
economic value that depends on the probability forecast π̂t+1 is given by:

E [υt+1(π̂t+1, st+1) | Ωt] = bt+1 (πt+1 − qt+1) 1(π̂t+1 > qt+1). (28)



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 47

Suppose we have a set of probability forecasts and states for t = 1, . . . , T , then the ex-
pectation in (28) can be evaluated by averaging over the observations to give the average
realized economic value:

υ =
1

T

T∑
t=1

bt (st − qt) 1 (π̂t > qt) , (29)

where πt+1 is replaced by the binary indicator st, given that the true probabilities are not
observed.

The Kuipers score (Ks) is defined as:

Ks = H − F

where H is the ‘hit rate’, the proportion of the total number of Bad states that were
correctly forecast, and F is the ‘false alarm’ rate, defined as the proportion of the total
number of Good states that were incorrectly forecast as being Bad states. The advantage
of the Ks statistic over measures such as the quadratic and log probability scores (QPS
and LPS) is that always forecasting the Bad state to occur (or always forecasting the
Good state) will score zero, whereas such strategies may fare well on QPS and LPS. The
Ks evaluates forecasts of events rather than forecasts of the probabilities of events. The
Bad state is forecast to occur when 1 (π̂t+1 > qt+1) = 1. We can express H and F as:

H =

∑T
t=1 st1 (π̂t > qt)∑T

t=1 st
, F =

∑T
t=1 (1− st) 1 (π̂t > qt)∑T

t=1 (1− st)
.

[63] show that in some circumstances Ks, a nominally purely statistical evaluation criterion,
and the economic value criterion, are proportional to each other, that is:

υ = bs (1− s)Ks

where s = T−1
∑T

t=1 st, the estimate of the (unconditional) probability of the Bad state.
To obtain this expression, the decision problem has to be simplified by assuming that
bt = b, all t, qt = q = s, all t.

[14] discusses the application of the decision-based approach to the UK Monetary
Policy Committee’s forecasts of inflation.

5. Forecasting White Noise

Granger (1983) (see [50]) shows that white noise processes can be forecastable. This
suggested that the Box-Jenkins ([7]) time-series modelling approach might not fully exploit
the predictability of the time series. The Box-Jenkins approach consisted of fitting an
autoregressive-moving average (ARMA) model to the series yt where the AR and MA lag
polynomials were of high enough order to result in a white noise error term. That is, an
error term εt with no discernible structure, such that the conditional expectation is zero,



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 48

E (εt | It−1) = 0, where It = (yt, yt−1, . . .) (or, equivalently, It = (εt, εt−1, . . .)). At the
time, autocorrelation and partial autocorrelation functions were used to identify the form
of the model (that is, the orders of the AR and MA polynomials), and to diagnose any
systematic patterns in the model’s error terms, so the emphasis was on linear dependence
in the time series, and on any non-modelled linear dependence in the error term after
fitting the selected ARMA model.

Granger argued that modelling a time-series up to a white noise error may still leave
room for improvement in terms of forecasting, and argued for the development of new
techniques able to detect non-linear dependence.

We will illustrate with two examples taken from [50]. The first is a simple linear model
where the use of a wider information set than that suggested by the Box-Jenkins approach
would yield more accurate forecasts, and the second is an example of a non-linear model
generating more accurate forecasts.

In the first, the variable yt is generated by:

yt = xt−1 + εt

where xt and εt are independent white noise. All the autocovariances of yt are zero, so
that yt is white noise. For example, consider Cov(yt, yt−1):

Cov (yt, yt−1) = Cov (xt−1 + εt, xt−2 + εt−1)

= Cov (xt−1, xt−2 + εt−1) + Cov (εt, xt−2 + εt−1)

which is zero because Cov (xt, xt−s) = Cov (εt, εt−s) = 0 for all t and s, other than s = 0,
by virtue of x and ε being white noise, and because Cov (xt, εt−s) = 0 for all t and s by
virtue of x and ε being independent. Moreover, all correlations of yt are zero, and not just
the first.

Next, consider the expected squared forecast error when the information set It includes
xt, so that the optimal forecast is E (yt | It−1) = xt−1. Then:

E
[
(yt − E (yt | It−1))2

]
= E

(
ε2t
)

= σ2ε .

Given that yt is white noise, the appropriate ARMA model is an ARMA(0, 0), i.e., yt = ut,
where ut is a white noise error. For this model, the information set is the null set,
E (yt | ∅) = 0, and the forecast-error variance is:

E
(
u2t
)

= σ2ε + E
(
x2t−1

)
.

Hence the population ARMA model’s forecasts are inferior to the model which conditions
on xt−1.

Granger defines the ‘usefulness’ of a forecasting model as:

R2 = 1− V ar(et)

V ar (yt)



M. P. Clements / Eur. J. Pure Appl. Math, 10 (1) (2017), 30-57 49

where et = yt −E (yt | It−1), and in this example, 0 < R2 < 1, whereas the equivalent R2

for the ARMA(0, 0) is zero.
[25] discuss the role of information in economic forecasting, and establish that ‘more in-

formation unambiguously does not worsen predictability, even in intrinsically non-stationary
processes’, where a variable is said to be predictable ‘if its distribution conditional on It−1
differs from its unconditional distribution’. Hence their definition of predictability corre-
sponds to Granger’s notion of ‘usefulness’. For a stationary process, V ar (et) = V ar (yt)
when the forecast (the conditional expectation) is equal to the unconditional mean, since

E (yt | It−1) = E (yt) implies V ar (et) = E
[
(yt − E (yt | It−1))2

]
= E

[
(yt − E (yt))

2
]

=

V ar (yt). [25] go on to further refine their notions of predictability and forecastability
when models are mis-specified and there is non-constancy (that is, given the worldview of
[23, 24]).

The second example is a bilinear model (Granger and Andersen (1978), see [55]):

yt = βyt−2εt−1 + εt

One can show that 0 < R2 < 0.5 when the model is invertible (βσε < 0.707), so that
although yt appears to be white noise it is forecastable non-linearly. Related examples are
discussed, including:

yt = βxt−1yt−1 + ut (30)

where xt and ut are Gaussian white noise. Then it follows that the autocorrelation function
of yt is everywhere zero, indicating white noise, and moreover Cov (yt, xt−s) = 0 for all s,
indicating that yt cannot be forecast linearly from xt. However, as indicated by (30), y is
forecastable from past values of y and x.

Granger and Teräsvirta (1993) (see [65]) provide an extensive treatment of non-linear
models and forecasting, and Granger’s work on non-linear models is the topic of the
contribution by Timo Teräsvirta.

6. Further Reading

This review does not aim to be comprehensive. Topics which are not covered include
the following. The concept of ‘time distance’ as an alternative to the standard approach
to evaluating forecasts based on the ‘vertical distance’, see: Granger and Jeon (1992,2003),
[67, 68]. The forecasting of transformed series, in Granger and Newbold (1976) (see [61]),
and of the Box-Cox transformation, in Nelson and Granger (1979) (see [84]). Finally, we
have not discussed Granger’s work on stock market price predictability and the efficient
markets hypothesis (Granger and Morgenstern (1970), [57], Timmermann and Granger
(2004), [97] and Granger (1992), [66]).

[77] provides a readable account of Granger’s work, not confined to forecasting.



REFERENCES 50

Acknowledgements

Michael Clements is Professor of Econometrics at the ICMA Centre, Henley Business
School, University of Reading, and an associate member of the Institute for New Economic
Thinking at the Oxford Martin School, University of Oxford. I am grateful to Jennifer
Castle and David Hendry for helpful comments and suggestions.

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