1 Model error and its estimation with particular application to loss reserving

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Model error and its estimation, with

particular application to loss reserving

By Greg Taylor

School of Risk and Actuarial Studies, University of New South Wales, Randwick,

AUSTRALIA

and

Gráinne McGuire

Taylor Fry, Sydney, AUSTRALIA

September 2022

Abstract. This paper is concerned with forecast error, particularly in relation to loss reserving.

This is generally regarded as consisting of three components, namely parameter, process and

model errors. The first two of these components, and their estimation, are well understood, but

less so model error. Model error itself is considered in two parts: one part that is capable of

estimation from past data (internal model error), and another part that is not (external model

error). Attention is focused here on internal model error.

Estimation of this error component is approached by means of Bayesian model averaging,

using the Bayesian interpretation of the LASSO. This is used to generate a set of admissible

models, each with its prior probability and the likelihood of observed data. A posterior on the

model set, conditional on the data, results, and an estimate of model error (contained in a loss

reserve) is obtained as the variance of the loss reserve according to this posterior.

The population of models entering materially into the support of the posterior may turn out to

be “thinner” than desired, and bootstrapping of the LASSO is used to gain bulk. This provides

the bonus of an estimate of parameter error also. It turns out that the estimates of parameter

and model errors are entangled, and dissociation of them is at least difficult, and possibly not

even meaningful. These matters are discussed.

The majority of the discussion applies to forecasting generally, but numerical illustration of the

concepts is given in relation to insurance data and the problem of insurance loss reserving.

Keywords. Bayesian model averaging, bootstrap, bootstrap matrix, forecast error, GLM,

internal model structure error, LASSO, loss reserving, model error.

1. Introduction

This paper is concerned with forecast error, particularly in relation to loss reserving. The

majority of the concepts are fairly general, and might be applied in many forecasting

environments, but the illustrations given here relate to loss reserving. To an extent, the paper

is a sequel to McGuire, Taylor and Miller (2021), which dealt with estimation of a loss reserve

by means of the LASSO. Both papers make applications of the LASSO, the first paper to point

estimation, and the sequel to estimation of forecast error, particularly model error.

When a forecast is made on the basis of a model of observations, it will inevitably contain an

error relative to the true value yet to be observed. Estimation of the properties of this error will

provide some understanding of the reliability of the forecast. This has been an issue in the loss

reserving literature since raised by Reid (1978), De Jong and Zehnwirth (1983) and Taylor and

Ashe (1983).

In subsequent years, forecast error has been decomposed into a number of components, most

notably parameter, process and model errors (see e.g. Taylor (1988), O’Dowd, Smith and

Hardy (2005), Taylor and McGuire (2016), Taylor (2021)). Estimation of the first two of these

three has become well understood, but there has been little development of the estimation of

model error.

A notable exception was O’Dowd, Smith and Hardy (2005) and Risk Margins Task Force

(2008), who provided a framework for the estimation of each component of forecast error using

scorecards to score subjectively a range of factors identified as likely to influence the quantum

of forecast error.

Where objective estimates are concerned, Taylor (2021) investigated model distribution error,

a component of model error and Bignozzi and Tsanakas (2015) consider model risk in the

context of VaR estimation, which is of course relevant to loss reserving, but loss reserving

models as such are beyond their scope. Blanchet, Lam, Tang and Yuan (2019) estimate the

effect of model error on a performance statistic in terms of the extrema of that measure over

the set of admissible models.

However, to the authors’ knowledge, there has been no other progress in the actuarial loss

reserving literature.

In the meantime, the subject has been addressed elsewhere in the economics and finance

literature. Useful general overviews are given by Glasserman and Xu (2014) and Schneider

and Schweizer (2015) in the context of financial risk management. The approach of Blanchet,

Lam, Tang and Yuan (2019) is similar to the latter.

Huang, Lam and Tang (2021) estimate the total of parameter error and model error (they call

these data variability and procedural variability respectively) in relation to deep neural

networks. Their approach equates more or less to bootstrapping, but where the replications are

obtained by random variation of the network initialization rather than data re-sampling.

The literature gives certain prominence to Bayesian model averaging (“BMA”) (Raftery, 1995,

1996; Raftery, Madigan and Hoeting, 1997; Hoeting, Madigan, Raftery, and Volinsky,1999;

Clyde and George, 2004; Clyde and Ivesen, 2013), and model confidence sets have also been

considered (Hansen, Lunde and Nason, 2011, and others).

3 
 
An example of this approach appears in the econometric literature in Loaiza-Maya, Martin and 
Frazier (2021), whose focus Bayesian prediction is similar to the Bayesian approach followed 
in the present paper, but with conditional likelihood replaced by a scoring rule. Martin, Loaiza-
Maya, Maneesoonthorn, Frazier and Ramírez-Hassan (2021) give a non-Bayesian presentation 
of the same ideas. 
This literature has been highly valuable at the fundamental conceptual level. However, the 
concepts are not easy to operationalize, and the much of the literature does not address loss 
reserving specifically.  
The present paper endeavours to fill some of the literature gaps identified above. It uses the 
LASSO (Hastie, Tibshirani and Friedman, 2009) to populate abstract concepts, such as model 
set and its prior distribution, that occur in the more theoretical literature. Since the LASSO may 
also be used as the source of a loss reserving model, this creates a direct nexus between that 
model and the estimation of its model error.  
The result is that, using the procedures described here, one may perform the following entire 
sequence of operations: 
• model a data set of claim observations and extract a point estimate of loss reserve; 
• move on to estimating the distributions of several components of forecast error that 
constitute a major part of the total; 
• supplement these with the distributions of the missing components, derived from other 
sources; 
• apply these distributions to the calculation of loss reserve risk margins, or any other 
quantities of interest that depend on the distribution of forecast error. 
The  paper  is  arranged  as  follows.  After  the  establishment  of  the  necessary  mathematical 
framework  and  notation  in  Section  2,  the  structure  of  the  problem  to  be  considered  is 
established with a review of the components of forecast error, and thereafter the paper focuses 
on one particular component, internal model structure error (“IMSE”) (Section 3). Section 
4 discusses the estimation of IMSE in the abstract, and the ingredients required for it. Then 
Section 5 puts these concepts to work in the specific context of the LASSO. This derives an 
estimate of the distribution of IMSE, but this estimate is then strengthened by bootstrapping 
the LASSO in Section 6. The whole procedure is then applied to several synthetic data sets, of 
varying complexity, in the derivation  of numerical results in Section 7. Finally, Section 8 
summarizes and considers the successes and limitations of the paper in the attainment of its 
objectives.  
 
2. Reserving framework and notation 
As far as possible, the notation here will follow that of McGuire, Taylor and Miller (2021).  
Accordingly, the analysis below will be concerned with the conventional claim triangle.  Some 
random variable of interest  is labelled by accident period  and development 
period .  In this setup, a cell of the triangle refers to the combination , 
and the observation  in this cell denoted . The payment period to which cell  relates 
will be denoted by . 

Let  denote the collection of cells (i.e. ordered pairs ) of which the triangle consists, and

let  denote the observations on these cells, i.e. . Accident and

development periods will be assumed of equal duration, but not necessarily years. As further

notation, 

. A realization of  will be denoted .

Let  be any random vector defined on  and  its realization. It will sometimes be useful

to vectorize these quantities, and so  will denote the column vector of all  listed in some

defined order, and  the corresponding vector of all .

This paper will be concerned with forecasts produced by models fitted to the data . Forecasts

are made in respect of the  for , some set of cells, disjoint from , and relating to

the future, i.e. . These  will now be denoted 

, and the vector of these will be

.

The forecast of 

 by model  will be denoted 



 and is equal to  for some real-

valued function . As the notation indicates the forecast is -dependent but, as this notation

is cumbersome, the  will be suppressed and the forecast written as simply 



 when this does

not create any ambiguity. Other quantities derived from 



 (e.g. immediately below) will also

be notated without explicit mention of .

A model  will include a likelihood function  for the observations so that the likelihood

of the data vector  is

. Let , the log- likelihood function.

It will be convenient to denote a value fitted by the model to a past observation  by  

. According to the vector notation given above,  denotes the vector of  for 

. Similarly, let 

 denote the vector of 

 for .

The modeller may select the model  but, in practice, will rarely know whether it is a correct

representation of the data. Therefore, assume that  is selected from some collection  of

candidate models, hereafter called the model set. Assume further that the model set is equipped

with a measure .

Suppose that , for some real-valued function , and define its forecast as 





. An example is , where 1 is a vector of the same dimension as  and with all

components equal to unity. If the 

 denote claim payments, then  is the amount of

outstanding claim liability.

The forecast error associated with forecast 

 will be defined as





(2.1)

Later sections will make use of open-ended ramp functions. These are single-knot linear

splines with zero gradient in the left-hand segment and unit gradient in the right-hand segment.

In a machine learning context, one of these would be referred to as a rectified linear unit

(“ReLu”). Let  denote the open-ended ramp function with knot at . Then

.

(2.2)

For a given condition , define the indicator function  when  is true, and 

when  is false. Further, define the Heaviside function .

3. Components of forecast error

Some regulatory regimes require that a capital margin be associated with a technical reserve

such that the total of reserve plus margin equal at least the Value at Risk (“VaR”) of the

liability at some high percentile, such as 99.5%. These regimes may also require the evaluation

of a risk margin within the capital margin. For example, the Australian prudential standards

require a loss reserve to be at least equal to the 75% VaR of the associated liability (Australian

Prudential Regulatory Authority, 2018).

IFRS17 requires a risk adjustment for non-financial risk. While there is no prescribed method

for calculation, margins based on the Value at Risk (“VaR”) are likely to be widely used.

For the sake of definiteness, this paper proceeds on the basis that the specific liability forecast

under consideration is the loss reserve. For some regulatory regimes (e.g. IFRS17) it might be

the one-year-ahead forecast of claim payments, with an associated VaR. In such cases, the

model error methodology set out in subsequent sections translates readily to this alternative

situation.

The requirement of a VaR necessitates the estimation of the distribution of a forecast liability,

as opposed to a simple point estimate. Often, especially in the case of low- or medium-

percentile VaRs, it is reasonable to assume that the required distribution is characterized by its

mean (the point estimate) and variance. Hence the need for examination of the variance of

forecast error.

It consists of a number of identifiable components. Decomposition of forecast error is discussed

by Taylor (2000), Taylor and McGuire (2016), McGuire, Taylor and Miller (2021), Taylor

(2021), O’Dowd, Smith and Hardy (2005), Risk Margins Task Force (2008) and Hastie,

Tibshirani and Friedman (2009), Huang, Lam and Zhang (2021), among others.

The different authors use slightly different terminology, and so Table 3-1 displays the

correspondences between the different terminologies. A blank entry indicates that the

component concerned is not explicitly considered by the authors in question. Correspondences

are sometimes exact, but at other times are a little rough because of different approaches taken

by different authors.

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1Modelerroranditsestimation,withparticularapplicationtolossreservingByGregTaylorSchoolofRiskandActuarialStudies,UniversityofNewSouthWales,Randwick,AUSTRALIAandGráinneMcGuireTaylorFry,Sydney,AUSTRALIASeptember2022Abstract.Thispaperisconcernedwithforecasterror,particularlyinrelationtolossreserving.Thi...

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