#### 2.1. Models and Econometric Approach

As noted earlier, we build on the portfolio time series approach for testing risk factor sensitivities evident in the literature (notably,

Fama and French 1993,

2012). A list of variables and their definitions are provided in

Appendix A. Energy utility portfolio returns are denoted in the generalised form

${\mathit{R}}_{i,t}$, where

${\mathit{R}}_{i,t}$ denotes the excess return

2 over a one-month UK treasury bill for the

$i$th portfolio on day

$t$. The econometric modelling begins with, respectively, the unconditional global AFFM and the local AFFM specifications, estimated using ordinary least squares regressions:

where

${\alpha}_{i}$ denotes the intercept,

${b}_{i}$ denotes the market coefficient,

${\mathit{R}}_{m,t}$ denotes the excess return on the market factor over the one-month UK treasury bill at time

$t$,

${s}_{i}$ denotes the

$SMB$ coefficient,

$SM{B}_{t}$ (

$LSM{B}_{t}$) denotes the global (local) size factor at time

$t$,

${h}_{i}$ denotes the

$HML$ coefficient,

$HM{L}_{t}$ $\left(LHM{L}_{t}\right)$ denotes the global (local) value factor at time

$t$,

${m}_{i}$ denotes the

$UMD$ coefficient,

$UM{D}_{t}$ (

$LUM{D}_{t})$ denotes the global (local) momentum factor at time

$t$,

$t{p}_{i}$ denotes the term premium coefficient,

${\mathit{R}}_{tp,t}$ denotes the term premium

3 at time

$t$,

${o}_{i}$ denotes the oil price risk coefficient,

${R}_{o,t}$ denotes the return on oil price at time

$t$,

${c}_{i}$ denotes the coal price risk coefficient,

${R}_{c,t}$ denotes the return on coal price at time

$t$,

${g}_{i}$ denotes the natural gas price risk coefficient and

${R}_{g,t}$ denotes the return on natural gas price at time

$t$. We include term premium and commodity risk factors based on empirical evidence of their significance in explaining oil industry and energy utility returns (

El-Sharif et al. 2005;

Koch and Bassen 2013;

Oberndorfer 2009;

Sadorsky 2001) or stock returns generally (

Fama and French 1993;

Batten et al. 2017;

Smyth and Narayan 2018).

Despite superficially similar model specifications, there are major differences between the global and local AFFMs. The global AFFM in Equation (1) uses global stock market risk factors calculated across a diversified sample of 600 European stocks as independent variables, with the objective of creating an integrated global AFFM which can be applied across sectors; i.e.,

$SM{B}_{t}$,

$HM{L}_{t}$ and

$UM{D}_{t}$ calculations. In contrast, the local AFFM in Equation (2) uses local stock market risk factors calculated across a sample of 88 European energy utilities as the independent variables, with the objective of more accurately explaining within-sector returns; i.e.,

$LSM{B}_{t}$,

$LHM{L}_{t}$ and

$LUM{D}_{t}$. For the third analytical focus, the analysis of time-varying risk factors, the local AFFM specification is applied in the following conditional regressions:

where the variables are the same as in Equation (2) (see

Section 2.1) save for the addition of

$co{2}_{i}$, which denotes the carbon price risk coefficient, and

${R}_{co2}$, which denotes the return on carbon

4.

The regressions are estimated using

Newey and West (

1987) heteroskedastic and autocorrelated consistent (HAC) standard errors, and subject to standard regression diagnostic tests.

#### 2.4. The 12 Energy Utility Portfolios

Beyond examining average returns for the energy sector as a whole, the 88 European energy utilities are also sorted into various portfolios based on the similarity of characteristics. The groupings, outlined in the following paragraphs, produce 12 portfolios to be examined: the energy sector as a whole, two portfolios based on size, three portfolios based on BE/ME ratios, three portfolios based on momentum and three portfolios based on industry.

The value-weighted returns of the 12 portfolios become dependent variables in the local AFFM in Equation (2) and the three ancillary asset pricing models: CAPM, augmented-CAPM and a local four-factor model. The purpose of the portfolio approach is to examine the within-sector heterogeneity of energy utility returns based on company characteristics. The benefit of this approach is the ability to examine the risk exposure of particular groups of utilities in isolation, for example, the risk exposure of small utilities.

First, we construct two stock portfolios based on company size. At the end of June of each year

$t$, from 1996 to 2013, all energy utility stocks are ranked on market capitalisation to proxy for size. Annually, the median market capitalisation is used as the breakpoint to put stocks into two portfolios: small or big energy utilities. Value-weighted returns are calculated for both the small and big portfolios from July of year

$t$ to end of June for

$t+1$, denoted as

${\mathit{R}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}}$ and

${\mathit{R}}_{\mathit{b}\mathit{i}\mathit{g}}$. The portfolios are rebalanced annually at the end of June for

$t+1$. Visual inspection showed that the two portfolios were well balanced each year, containing approximately equal numbers of stocks. The median number of stocks in the

${\mathit{R}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}}$ and

${\mathit{R}}_{\mathit{b}\mathit{i}\mathit{g}}$ portfolios, across all years, is 22.5. Although balanced, big energy utilities naturally dominate sector valuation—the combined value of small energy utilities account for 6.4% of total sector valuation; this is consistent with the global AFFM and

Fama and French (

1995). For the global AFFM, small stocks account for 5.84% of the total market value, across all stocks and years, while for

Fama and French (

1995), small stocks accounted for about 7.3% of total market value in 1991. The

${\mathit{R}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}}$ and

${\mathit{R}}_{\mathit{b}\mathit{i}\mathit{g}}$ portfolios will be used as dependent variables in Equation (2) to examine heterogeneous risk exposure based on utility size.

To form the three BE/ME portfolios, all energy utilities are ranked on their BE/ME ratios annually. The BE/ME ratio is calculated as the book value of common equity for the fiscal year ending in calendar year

t − 1, scaled by market capitalisation at the end of December in year

t − 1. The energy utilities are allocated to groups based on

Fama and French’s (

1993,

1995,

1997,

1998,

2006,

2012) three breakpoints: the top 30% (high-BE/ME), the middle 40% (mid-BE/ME) and the bottom 30% (low-BE/ME). The three groups represent value, neutral and growth stocks, respectively (

Fama and French 2006,

2012;

French 2015). There were only two negative BE/ME calculations, which were excluded from the portfolio. The high-, mid- and low-BE/ME portfolios contain a median of 13, 18 and 13.5 companies, respectively, across all years. Value-weighted returns are calculated for the high-BE/ME, mid-BE/ME and low-BE/ME portfolios, denoted as

${\mathit{R}}_{\mathit{h}\mathit{i}\mathit{g}\mathit{h}}$**,** ${\mathit{R}}_{\mathit{m}\mathit{i}\mathit{d}}$ and

${\mathit{R}}_{\mathit{l}\mathit{o}\mathit{w}}$, respectively. The portfolios are rebalanced at the end of June for

$t+1$. The three portfolios will be used as dependent variables in Equation (2) to examine heterogeneous risk exposure based on the book-to-market ratio.

9To form the three momentum portfolios, the average excess return for all 88 European energy utilities is calculated daily over the formation period from day

$t-251$ to day

$t-21,$ and excludes the sort month. To be considered as an up-momentum utility, the energy stock’s returns during the formation period and on

$t-21$ must be positive; similarly, the stock returns during the formation period and return on

$t-21$ must be negative for down-momentum utilities. The

$t-21$ condition ensures that the up and down momentums continue until the end of the formation period, and reversal has not already begun. The daily breakpoints are defined as the top 30% (up-momentum), the middle 40% (neutral-momentum) and the bottom 30% (down-momentum). The median number of stocks in the three momentum portfolios, across all years, is 13, 18, and 13, respectively. The value-weighted daily returns on the up, neutral and down momentum portfolios are calculated, rebalanced daily and denoted as

${\mathit{R}}_{\mathit{u}\mathit{p}}$,

${\mathit{R}}_{\mathit{n}\mathit{e}\mathit{u}\mathit{t}\mathit{r}\mathit{a}\mathit{l}}$ and

${\mathit{R}}_{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}$, respectively. The

${\mathit{R}}_{\mathit{u}\mathit{p}}$,

${\mathit{R}}_{\mathit{n}\mathit{e}\mathit{u}\mathit{t}\mathit{r}\mathit{a}\mathit{l}}$ and

${\mathit{R}}_{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}$ portfolios will be used as dependent variables in Equation (2) to identify whether the risk factors for energy utilities differ based on momentum. Based on

Moskowitz and Grinblatt (

1999),

Boni and Womack (

2006) and

Fama and French (

2012), the three momentum portfolios are expected, by definition, to have extreme momentum tilt, and thus the local AFFM may have difficulty capturing average returns.

To form the three industry portfolios (electricity, natural gas or multi-utility), we obtain up to 10 SICs for each energy utility annually between 1996 and 2013

10. We group the companies into portfolios based on their SICs. The SIC system is designed to categorise industries using a four-digit code. Grouping companies by SICs is similar to the approach employed by

Moskowitz and Grinblatt (

1999).

Based on the SICs, companies that exclusively contained only electricity and “other” operations are defined as electric utilities, companies which contained only natural gas and “other” operations are defined as natural gas utilities, and companies which contained operations from both electricity and natural gas, or were otherwise defined as multi-utilities, are defined as multi-utilities. Auxiliary operations outside the electricity sector are minor and are not expected to significantly impact returns.

As SIC codes define the business operations which generate the highest revenue for the companies in the past year (

$t)$, SIC codes for year

$t$ are matched

11 to returns for July of year

$t$ to June of

$t+1$. The value-weighted daily returns on the electricity, natural gas and multi-utility portfolios are calculated, denoted as

${\mathit{R}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\mathit{u}\mathit{t}\mathit{i}\mathit{l}}$,

${\mathit{R}}_{\mathit{g}\mathit{a}\mathit{s}\mathit{u}\mathit{t}\mathit{i}\mathit{l}}$ and

${\mathit{R}}_{\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}}$, respectively. The number of stocks in the three industry portfolios, across all years, is 24, 7, and 14, respectively. The portfolios are rebalanced annually in June of year

$t+1$ to control for utilities that change operations or industries. We do so to control for company mergers, where the acquiring company shifts operations from, say, electricity to multi-utility operations. Although rare, some of the SICs of utilities have changed across the years but were mostly confined to ancillary operations rather than primary operations.

The 12 portfolios defined above are used as dependent variables for analysis in Equation (2), where ${\mathit{R}}_{i,t}\equiv $ ${\mathit{R}}_{\mathit{u}\mathit{t}\mathit{i}\mathit{l},\mathit{t}}$, ${\mathit{R}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l},\mathit{t}}$, ${\mathit{R}}_{\mathit{b}\mathit{i}\mathit{g},\mathit{t}}$, ${\mathit{R}}_{\mathit{h}\mathit{i}\mathit{g}\mathit{h},\mathit{t}}$, ${\mathit{R}}_{\mathit{m}\mathit{i}\mathit{d},\mathit{t}}$, ${\mathit{R}}_{\mathit{l}\mathit{o}\mathit{w},\mathit{t}\text{}}$,$\text{}{\mathit{R}}_{\mathit{u}\mathit{p},\mathit{t}}$, ${\mathit{R}}_{\mathit{n}\mathit{e}\mathit{u}\mathit{t}\mathit{r}\mathit{a}\mathit{l},\mathit{t}}$,$\text{}{\mathit{R}}_{\mathit{d}\mathit{o}\mathit{w}\mathit{n},\mathit{t}}$ ${\mathit{R}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\mathit{u}\mathit{t}\mathit{i}\mathit{l},\mathit{t}}$,$\text{}{\mathit{R}}_{\mathit{g}\mathit{a}\mathit{s}\mathit{u}\mathit{t}\mathit{i}\mathit{l},\mathit{t}}$ or ${\mathit{R}}_{\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i},\mathit{t}}$. Each portfolio is regressed independently. The following section explains the construction of the local stock market risk factors used as independent variables in Equation (2).

#### 2.5. Time-Varying Risk Factor Sensitivities

To address the third analytical focus, relating to time-varying risk factor sensitivities (stage one of the analysis) involves implementing conditional annual regressions in Equation (3) to account for slope shifts over time, as has been the norm in the literature (

Tulloch et al. 2017a;

Batten et al. 2017;

El-Sharif et al. 2005). We compare the results to those obtained from the

Bai and Perron (

2003) sequential multiple breakpoint models.

In the second part of the analysis, we employ the

Bai and Perron (

1998,

2003) inductive structural breakpoint algorithm to examine the presence of multiple structural changes in model parameters. The inductive approach can overcome many of the misspecification criticisms of the deductive approach (our annual regressions in stage one), such as assumptions regarding the break date, and allows the examination of structural change where the breakpoint is entirely unknown. The

Bai and Perron (

1998,

2003) algorithm utilises previously forgotten dynamic programming of pure structural change models for a more general partial structural change model, specifically, partitioned regressions and cluster analysis, curve fitting by use of segmented straight lines (polygonal curves) and grouping for maximum homogeneity by minimising variance within groups (see, respectively,

Guthery 1974;

Bellman and Roth 1969;

Fisher 1958). The

Bai and Perron (

1998,

2003) algorithm is implemented in two steps: (1) a posthoc multiple breakpoint test, and (2) the breakpoint regression, as explained below.

The first step implements posthoc parameter stability diagnostic tests on the results of the local AFFM in Equation (2). The multiple breakpoint test identifies whether there are potential breakpoints in the unconditional local AFFM’s parameters. The break specification is sequential, testing the null of $\ell $ versus the alternative of $\ell +1$ breaks. The information criterion is set to allow up to 18 structural breaks, the maximum available, and employs a trimming percentage of 5%. As the dataset consists of 4435 observations, the trimming value implies that regimes must have at least 222 observations to be considered a structural break; this was the minimum period permissible by the model. The significance level is $p\le 10\%$, and error distributions are allowed to differ across breaks to control for heterogeneity across time periods. The results of the test report an estimate for the number of potential breaks in the sample and the estimated break dates.

The second step implements a breakpoint regression, specifying the local AFFM (Equation (2)) as the mean equation. The breakpoint regression estimates a linear regression where the parameters are subject to structural change. The algorithm obtains global minimisers of the sum of squared residuals (SSR) based on dynamic programming. Based on the evidence of heteroskedasticity and autocorrelation, Newey-West HAC standard errors for the coefficient covariance matrix are used, and error distribution is allowed to differ across breaks to account for the heterogeneity of time periods. The results of

Bai and Perron (

2003) show that this allowed for the detection of smaller breaks, which were otherwise obscured in the data. The HAC coefficient covariance matrix automatically determines optimised lag structuring using the Akaike Information Criterion

12 (AIC). The kernel bandwidth is automatically determined using Andrew’s autoregressive method with 1 lag (AR(1)) and uses quadratic-spectral kernels. To remain congruent with the first stage, the break specification is also sequential, testing the null of

$\ell $ versus the alternative of

$\ell +1$ breaks. Again, the information criterion is also set to allow a maximum of 18 structural breaks, employing the same trimming percentage of 5% and test significance at

$p\le 10\%$. The test will estimate the date of structural breaks in the relationship between returns in the energy sector and the risk factors of the local AFFM. The results also report the estimated coefficients across each of the break dates, allowing examination of the changing relationship with risk factors through time.