Modelling Sector-Level Asset Prices

We present a modelling approach for sector asset pricing studies that incorporates sector-level risk factors, sub-group portfolios, and structural break point tests that are better at isolating the time-varying nature and the firm-specific component of returns. The sub-group portfolios show considerable subsector heterogeneity, while the asset pricing model using local risk factors and inductive structural breaks results in a superior model (R2 of 80.42% relative to R2 of 68.79% of ‘conventional’ models). Finally, we show that 28% of the variance of residuals, normally assumed to be the firm-specific component of returns, can be attributed to the changing relationship between sector returns and risk premia.


Introduction
This paper outlines a modelling approach for implementing asset pricing studies at the sector level.
We show how enhanced understanding of sector returns and risk factors can be achieved through four foci: 1) by calculating stock market risk factors at the sector level; 2) by creating sub-group portfolios to explore within-sector heterogeneity; 3) by applying inductive structural break point tests to identify time-varying risk premia; and 4) by better isolating the firm-specific component of returns. We do so in the context of the European energy utility sector which has undergone dramatic changes over the last two decades, making it an ideal context to explore local and time-varying risk factors as well as within-sector heterogeneity. Fama and French (1993) document that sector-level peculiarities with regard to stock market risk factors can be a major influence in capturing variation in stock returns. Fama and French (1997) explore the sector-level performance of the capital asset pricing model (CAPM) and the three-factor model, employing both models on 48 U.S. industries between 1963 and 1994. They find that the choice between the two models can result in large differences in the valuation of investments; the cost of equity calculation differs by more than 2% for 17 industries and more than 3% for eight industries. Fama and French (1997) argue that discrepancies in the cost of equity estimates at the sector level are partly caused by estimation error in two risk factors: size and value premia. The cause for the large estimation error arises from the return profiles of a sector differing to that of the market as a whole.
Further motivation for examining returns using sector-level data come from Boni and Womack (2006). Their results show that a sector-based momentum strategy substantially improves the returns relative to risk borne. The overall conclusion is that sector-specific analysts are good stock pickers within their sector of expertise, and investors acknowledge that the analysts' information is valuable with respect to identifying within-sector mispricing. These results imply that within-sector characteristics need to be accounted for in sector analyses (Boni and Womack 2006).
The typical approach to building an asset pricing model is to use market-level (global) stock market risk factors. Global risk factors capture a wide range of variation in returns across a variety of different stocks in the same market. For example, a researcher may be interested in explaining the average returns of European energy utilities against a broad portfolio of European stocks. But for a more nuanced understanding of the average returns within a sector, it is preferable to use sector-level (local) risk factors. Local risk factors ensure that only the variations in return which are relevant to the sector of interest is retained in the asset pricing model, for example, to explain the average returns of small energy utility stocks relative to the average returns of energy utility stocks. Stocks from other sectors can contribute little to explaining the variation in returns. sectors as calculating the stock market risk factors at the sector level reduces dispersion in the unconditional mean for the risk factors and provides a more accurate measurement of the impact of size, value and momentum on average returns. Accordingly, this paper implements an 'augmented' local four-factor model (AFFM) on the European energy utility sector. Our model is augmented in that it includes additional commodity and macroeconomic risk factors used in the energy economics literature (see Tulloch, Diaz-Rainey, and Premachandra 2017a).
In the second part of our analysis, we take the analysis one step further by examining withinsector heterogeneity. We do so by implementing the local AFFM in the context of 12 energy portfolios: the energy sector as a whole, two portfolios based on size, three portfolios based on bookto-market (BE/ME) ratios, three portfolios based on momentum and three portfolios based on industry (sub-sector).
The third and fourth parts of the analysis explore time-varying risk premia and the firmspecific component of return. Structural breaks in a time series can cause pure and partial changes in model parameters affecting the firm-specific component returns after filtering out systematic components (Fama and French 1993;Hansen 2001). Accordingly, we control for this by applying a Bai and Perron (1998; inductive structural break point modelling approach. We adopt the inductive approach because the deductive approach to testing break points can lead to biased significance tests, unless the breaks are known with certainty. Surplus observations which are unaffected by structural breaks skew the mean residuals towards zero, reducing the power of statistical tests and biasing significance tests (Quandt 1960).
In terms of time-varying risk premia, the Bai and Perron (1998; tests are utilised in two ways: 1) as a post-hoc stability diagnostic tool and 2) as a break point regression. In the final part of our analysis, we introduce a novel approach to better isolate the firm-specific component of returns.
This includes both conditional annual regressions and an inductive break point regression to construct an orthogonalised firm-specific return series.
As noted previously, the analysis is conducted in the context of the European energy utility sector. The EU's drive to create a single European energy market and the 'greening' of the energy supply has been described as the most extensive cross-border reform of energy networks and operating structures in the world (Tulloch, Diaz-Rainey, and Premachandra 2017a; Tulloch, Diaz-Rainey, and Premachandra 2017b; Jamasb and Pollitt 2005). As such, the EU energy sector is an ideal context to explore local and time-varying risk factors, as well as within-sector heterogeneity. We do so over the period 1996 to 2013, utilising a comprehensive sample of European energy utilities that controls for survivorship bias. 1 Our results show that local stock market risk factors explain a greater proportion of sector returns compared with global stock market risk factors. For the energy sector, the adjusted 2 increases from 68.79% using the global AFFM to 72.77% using the local AFFM.
Second, this paper identifies within-sector heterogeneity, examining the risk exposure of various energy utilities grouped on similarity of characteristics. Heterogeneous sensitivity to size, value and momentum premia were the largest determinants for the differences in expected returns across various energy portfolios. The overall results indicate that electricity utilities are riskier than the natural gas and multi-utility industries. Interestingly, the multi-utilities showed one of the lowest cumulative abnormal returns across all portfoliospossibly indicating a lower risk-return relationship.
The multi-utilities have less commodity risk exposure than both the natural gas and electricity industries. This is consistent with economy of scopediversified operations allow multi-utilities to switch operations when faced with regulatory changes or fluctuations in commodity prices.
Third, the inductive method of controlling for structural breaks improves the local AFFM's adjusted 2 to 80.42%. This is a large improvement in fit relative to using the conventional approach to asset pricing; full period (unconditional) regression produces adjusted 2 of 68.79% for the global AFFM and 72.77% for the local AFFM. This highlights the importance of controlling for time-varying risk premia, especially in the context of long time periods and sectors that have undergone large structural changes.
Finally, we introduce a better method to isolate the firm-specific component of returns and filter out systematic risk by using a Bai and Perron (1998; break point regression on the residuals of the original asset pricing regression. Our results show that, when using a conventional asset pricing model, almost 28% of the variance of residuals which is normally assumed to be the firm-specific component of returns can be attributed to the changing relationship between sector returns and risk premia.
Overall, this paper provides a template for conducting sector-level asset pricing studies which can be adapted to other sectors, since it suggests an approach of more accurately examining timevarying risk premia and isolating the firm-specific component of returns for any sector. This is important not just to investors but also regulators in regulated sectors who want to understand the impact of policies on sector risk-return dynamics and cost of capital.
The structure of the rest of this paper is as follows. Section 2 outlines the methodology and data; Section 3 presents the descriptive and econometric results; and Section 4 provides a concluding discussion. component of return. This paper is distinct yet complementary to Tulloch, Diaz-Rainey, and Premachandra (2017a) and Tulloch, Diaz-Rainey, and Premachandra (2017b).

Models and Econometric Approach
Energy utility portfolio returns are denoted in the generalised form , , where , denotes the excess return over the one-month UK treasury bill for the th portfolio on day . The econometric modelling begins with, respectively, the unconditional global AFFM and the local AFFM specifications, estimated using OLS regressions: where denotes the intercept, denotes the market coefficient, , denotes the excess return on the . For the annual conditional regressions, we use the local AFFM specification: where 2 denotes the carbon price risk coefficient and 2 denotes the return on carbon. 2 The regressions are estimated using Newey-West (1987) heteroskedastic and autocorrelated consistent (HAC) standard errors, and subject to standard regression diagnostic tests.

Sample and Data 3
Data were extracted from Thomson Reuters Datastream, S&P Capital IQ and publicly available sources. Data were measured in euros (€) to represent value and cost to European market participants.
The daily stock prices and market capitalisations of the energy utilities cover the period 30 June 1995 to 28 June 2013. 4 Stock prices are measured at day close and adjusted for corporate actions. Returns for all stocks and risk factors are calculated using the first-log difference. Excess returns for equities are calculated as the difference between daily returns and the daily yield on the one-month UK Treasury bill. Regarding the accounting data, all data are extracted for fiscal year-end, covering the period 1995 to 2013. To be eligible for analysis, and to allow portfolio rebalancing, all companies must have data on stock price, market capitalisation and book value of equity for both year and year − 1. This condition ensures companies have traded for at least two years (Fama and French 1993).
The STOXX® 600 Europe index ( , ) is used as a proxy for market returns, representing 600 large-, mid-, and small-capitalisation stocks across 18 countries of the EU. The inclusion of mid-and smallcap stocks prevents bias towards larger companies. The three global stock market risk factors of size ( ), value ( ) and momentum ( ) premia are calculated using all 600 stocks and the extensive portfolio method outlined by Fama and French (1993) and Carhart (1997), with annual portfolio rebalancing. The stock market risk factors are calculated using the same 600 stocks as the STOXX® 600 Europe index. 5 The momentum premium comprises the first year of daily data, so the analysis covers 01 July 1996 and 28 June 2013.
We construct a sample of 88 European energy utilities. The STOXX® 600 Europe Utilities index is used to provide an initial list of 28 utilities currently operating and traded on equity markets.
We remove all utilities whose primary revenue is derived from waste or water operations to prevent biased estimated coefficients. We extend the sample by including all companies explicitly mentioned in energy sector restructuring legislation or are elected members of various electricity and gas groups. 6 Using Standard Industrial Classification (SIC) codes, we control for survivorship bias by including all active and non-active energy utilities registered under the same product segments and SICs.

The Local Stock Market Risk Factors: Size, Value and Momentum Premia
While a diversified portfolio of 600 European stocks is used to create the global stock market risk factors, the local AFFM uses the 88 European energy utilities to create the local stock market risk factors. The sector-level-mimicking portfolios are used as independent variables in Equations 2 and 3.
The local (small minus big) risk factor ( ) mimics the risk factor in returns which is related to energy utility size, representing local size premium. The represents a zero-sum investment which is long on small energy utilities and short on big energy utilities. Small energy utilities are expected to generate higher returns than big energy utilities. The local (high minus low) risk factor ( ) mimics the risk factor in returns which is related to energy utility book-tomarket ratio. The represents a zero-sum investment which is long on high-BE/ME (value) energy utilities and short on low-BE/ME (growth) energy utilities, representing local value premium.
High-BE/ME utilities are expected to generate higher returns than low-BE/ME utilities. The local (up minus down) risk factor ( ) mimics the risk factor in returns which is related to energy utility persistence of earnings (momentum). The represents a zero-sum investment which is long on up momentum energy utilities and short on down momentum energy utilities and represents the momentum premium. Up momentum energy utilities are expected to generate higher returns than down momentum energy utilities.

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 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 , 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 break point 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 to end of June for + 1, denoted and . The portfolios are rebalanced annually at the end of June for + 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 and portfolios, across all years, is 22.5. Although balanced, big energy utilities naturally dominate sector valuationthe combined value of small energy utilities account for 6.4% of total sector valuation; however, 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 account for about 7.3% of total market value in 1991. The and 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 − 1, scaled by market capitalisation at the end of December in year − 1. The energy utilities are allocated to groups based on Fama and French's (1993;1995; 2012) three break points: 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;Fama and French 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 , and , respectively. The portfolios are rebalanced at the end of June for + 1. The three portfolios will be used as dependent variables in Equation (2) to examine heterogeneous risk exposure based on book-to-market ratio. 7 To form the three momentum portfolios, the average excess return for all 88 European energy utilities is calculated daily over the formation period from day − 251 to day − 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 − 21 must be positive; similarly, the stock returns during the formation period and return on − 21 must be negative for down momentum utilities. The − 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 break points are defined as the top 30% (up momentum), the middle 40% (neutral momentum) and the bottom 30% (down momentum). The value-weighted daily returns on the up, neutral and down momentum portfolios are calculated, rebalanced daily and denoted , and , respectively. The , and 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. 8 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 which 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 ( ), SIC codes for year are matched 9 to returns for July of year to June of + 1. The value-weighted daily returns on the electricity, natural gas and multi-utility portfolios are calculated, denoted , and , respectively. The portfolios are rebalanced annually in June of year + 1 to control for utilities which 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 changed across 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), , , , or , . 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).

Time-varying Risk Premia and Isolation of Firm-specific Returns
To address research foci 3, relating to time-varying risk premia, this paper employs the Bai and Perron (1998; inductive structural break point 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, such as assumptions regarding the break date, and can allow examination of structural change where the break point is entirely unknown. The Bai and Perron (1998; 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; algorithm is implemented in two stages: 1) a post-hoc multiple break point test and 2) the break point regression; explained below.
The first stage implements post-hoc parameter stability diagnostic tests on the results of the local AFFM in Equation (2). The multiple break point test identifies whether there are potential break points in the unconditional local AFFM's parameters. The break specification is sequential, testing the null of ℓ versus the alternative of ℓ + 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 4,435 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 ≤ 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. 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 heterogeneity of time periods. The results of Bai and Perron (2003) show that this allowed for 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 10 (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 ℓ versus the alternative of ℓ + 1 breaks. Again, the information criterion is also set to allow a maximum of 18 structural breaks, employs the same trimming percentage of 5% and tests significance at ≤ 10%. The test will estimate the date of structural breaks in the relationship between returns in the energy sector and the risk premia 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 premia through time.
To address analytics foci 4, we use the Bai and Perron (1998; algorithm to introduce a better method to isolate the firm-specific component of returns and filter out systematic risk. We posit that the assumption of constant risk factors in the asset pricing model forces any changes in the relationship between risk factors and returns into the residuals of an unconditional linear regression. We examine this proposition by extracting the residuals of three different modelling approaches and performing a 'second pass' test on the residual using the inductive Bai and Perron (1998; algorithm. The three modelling approaches include constant parameters using the unconditional local AFFM regression (Equation 2), time-varying parameters using conditional annual local AFFM regressions (Equation 3) and time-varying parameters of the local AFFM sequential break point regression. The 'second pass' examines whether the residuals are capturing any change in the relationship between the stock returns and the risk factors. If the modelling approaches adequately filter out systematic risk, leaving only the firm-specific component of returns, the residuals of the regression should have no relationship with the model parameters. We show this is not the case with the assumption of constant parameters. We compare the performance of the models and make a recommendation as to which approach better isolates firm-specific returns. Figure 1 (Plot A) compares the cumulative returns for the entire European energy utility sector and the two size portfolios: small and big energy utilities. As expected, the small energy utilities have higher cumulative returns than the big energy utilities, illustrating a clear size effect. Plot B of Figure 1 shows the cumulative return profiles for the three BE/ME portfolios. The high-BE/ME (value) energy utilities have higher cumulative returns than mid-BE/ME (neutral) energy utilities. The cumulative returns are consistent with the literature; high-BE/ME utilities are expected to outperform low-BE/ME utilities (Rosenberg, Reid, and Lanstein 1985;Chan, Hamao, and Lakonishok 1991;Fama and French 1992;Fama and French 1993;Fama and French 1995;Fama and French 1998). Plot C of Figure 1 illustrates the cumulative returns for the three momentum portfolios. Naturally, up momentum stocks outperform down momentum stocks. Finally, Plot D of Figure 1 presents the cumulative returns for the three industry portfolios. The results show that the electricity utility portfolio generated the greatest cumulative returns across time, indicating a higher risk-return relationship. In contrast, the natural gas and multi-utility portfolios show similar returns through time. The multi-utilities show one of the lowest cumulative abnormal returns across all portfolios. This is consistent with economy of scope. A diversified portfolio of operations is less likely to be exposed to the regulatory and operational risks of single utilities. Plot B of Figure 2 shows that the value premium, the spread between high-and low-BE/ME stocks, is greater in the energy sector compared with all European stocks. Finally, Plot C of Figure 2 shows a consistent momentum premium. Up momentum stocks outperform down momentum stocks for both all European stocks and within the energy utility sector. The important implications for the results above are that there are some differences between global and local stock market risk factors, especially with respect to the value premium. While the size and momentum premia are similar between the global and local risk factors, the cumulative return profiles are not identical.

Descriptive Statistics
The summary statistics for all 12 portfolios and 12 risk factors are presented in Table 1. Most of the returns across all portfolios and risk factors are not statistically different from zero, with the exception of the up momentum portfolio ( , ), the risk factor, risk factor, term premium and carbon risk. The summary statistics indicate that the mean daily return for the energy utility sector is -0.0051%, losing value over time. Small utilities achieve a greater mean return (0.0155%) compared with big utilities (-0.0062%), reflecting the greater risk-return relationship of small utilities. For momentum portfolios, the up momentum portfolio achieves a mean return of 0.0291%, significant at ≤ 0.01 and greater than the neutral and down momentum portfolios (0.0095% and -0.0022%, respectively). For the industry portfolios, the electricity and natural gas industries achieve mean returns of 0.0136% and 0.0127%, respectively. Multi-utilities only achieved a mean return of 0.0006%, consistent with the lower perceived risk and economy of scope argument presented above. Augmented Dickey-Fuller (1979) (ADF) unit root tests were implemented (not reported) to confirm the stationarity of the time series, ensuring the dependent and independent variables were integrated to the same order and that a linear relationship can exist between the variables. The results of the ADF test, confirm that the time series is integrated to order zero, (0), and stationary. To address assumptions of the linear regression, regression diagnostic tests identified heteroskedasticity and autocorrelation of residuals, which are reported in Table 2 and corrected for. All coefficients are estimated using the Newey-West (1987) HAC covariance matrices. The variance inflation factor statistics found no evidence of multicollinearity among variables.

Local Stock Market Risk Factors Better Explain Sector-level Returns
Addressing the first analytical focus, the use of local stock market risk factors captures a greater proportion of returns. The adjusted 2 for the energy sector as a wh ole is 72.77% using the local AFFM (Table 2, Model 4), compared with 68.79% using the global AFFM (Table 3, Model 5).
These results are congruent with Moskowitz and Grinblatt (1999) and Fama and French (2012). The local AFFM also produces the highest adjusted 2 in comparison with existing asset pricing models: the CAPM (66.96%), the augmented-CAPM used in the energy economics literature (67.17%) and the (local) four-factor model (72.56%).
Interpreting the results, the local AFFM (Model 4) shows that the sector is relatively defensive over the whole time period, with a market beta of 0.6306 ( ≤ 0.001). Further, the energy utility sector's returns covary with the returns on big energy utilities (they have a large negative slope on the factor), are marginally tilted towards behaving like low-BE/ME (growth) stocks (they have a small negative slope on the factor) and are tilted towards behaving like down momentum utilities (they have a marginally negative slope on the factor). Coal is the only statistically significant commodity which affects returns at the sector level, with a marginally negative slope.
It is informative to explore the differences between the estimated coefficients of the local AFFM and the global AFFM. The largest difference occurs in the slopes; the global AFFM shows that the energy utility sector behaved like high-BE/ME European stocks based on the global risk factors, typically associated with distressed companies. In contrast, the local AFFM shows that there is a slightly negative coefficient with the factor, suggesting a marginal tilt towards the low-BE/ME (growth) utilities.
Concluding the implications for the first analytical focus is that use of local stock market risk factors explains a greater proportion of returns at the sector level, a distinct improvement over the global stock market risk factors.

Portfolios within the Energy Sector Have Heterogeneous Risk Exposure
The second analytical foci of this paper is concerned with identifying within-sector heterogeneity. We do so by applying the local AFFM (Equation 2) on the 12 portfolios identified in Section 2.3. The results are also reported in Table 2 for ease of comparison.
The first observation regarding the energy portfolios is that it is rare for portfolios to experience extreme size, value or momentum tilt. Addressing the two size portfolios, the interpretation for big utilities is similar to the sector as a whole. Big utilities have a market beta of 0.640 and have a negative coefficient, which is expected. Big utilities are tilted towards behaving like low-BE/ME utilities (a negative ℎ coefficient) and towards down momentum (negative coefficient). Coal is the only significant commodity risk factor for big utilities, with a negative impact. The local AFFM typically performs poorly at explaining the returns on small energy utilities, with an adjusted 2 of 44.68%, consistent with the argument that smaller companies are typically harder to value and often informationally sparse (Kumar 2009). Overall, small energy utilities are still defensive investments, with a market beta of 0.494, but are more exposed stock market and commodity risk factors in comparison with big utilities. Naturally, the small utilities are expected to have a positive slope, but they also behave like high-BE/ME (value) utilities (positive ℎ coefficient) and are marginally tilted towards down momentum (negative coefficient). The positive size and value premia are consistent with utilities being distressed and/or marginal companies, requiring greater return on investment. Small utilities have positive oil risk and negative coal risk. The increased commodity risk exposure suggests that small utilities do not effectively hedge against commodity risk.
Regarding the three book-to-market portfolios, the mid-BE/ME (neutral) energy utilities have the greatest systematic risk, in comparison with high-BE/ME (value) and low-BE/ME (growth) utilities. The size premium suggests that the three portfolios behave like big energy utilities, while the momentum premium shows that the portfolios show some tilt towards down momentum. Interestingly, the high-BE/ME utilities, which are typically associated with company distress and fallen angels, show sensitivities to all commodities: oil, coal and natural gas risk. In contrast, the mid-and low-BE/ME portfolios show less commodity risk exposure. The significant commodity risk of the high-BE/ME and small utilities is consistent with Oberndorfer's (2009) and Kumar's (2009) propositions: commodities serve as informational signals for price developments in the energy sector when less information is available.
We do not draw inferences regarding the three momentum portfolios. Fama and French (2012) argue that local models have difficulty when capturing average returns for portfolios with extreme momentum tilt. We find similar results. Also, the momentum strategy does not represent a realistically viable investment opportunity, as an active portfolio strategy such as momentum requires extremely high turnover (Moskowitz and Grinblatt 1999). The trading costs of rebalancing the portfolio daily would have a negative impact on the strategy's performance (Rosenberg, Reid, and Lanstein 1985).
Note, none of the criticisms above affect the use of the risk factor when used to explain returns on other energy portfolios. In summary, the risk factor is useful as an independent variable but encounters known econometric issues as a dependent variable.
Turning to the three industry portfolios in Table 2, the electricity industry portfolio has the lowest market beta, is tilted towards big stock (negative coefficient), is marginally tilted towards low-BE/ME (negative ℎ coefficient) and marginally tilted towards down momentum (negative coefficient). The electricity industry has the greatest commodity risk exposure of all three industries; all commodities are statistically significant. Second, the natural gas industry has the highest market beta of the three industries, showing increased systematic risk. Natural gas utilities behave like big utilities (negative coefficient), are tilted towards low-BE/ME (growth) utilities (negative ℎ coefficient) and tilted towards up momentum (positive coefficient). The natural gas industry is the only industry with positive momentum, possibly indicating profiteering. Unsurprisingly, the natural gas sector has positive relationships with oil and natural gas prices. The implications for the sector are that they generally perform well relative to other utilities and have a larger market capitalisation.
Finally, the multi-utility sector, unsurprisingly, shares risk exposure with both electricity and natural gas utilities. Table 2 shows that the multi-utility portfolio shares similar market beta, momentum and coal price coefficients with the electricity industry, and similar size and value coefficients with the natural gas industry. This result is expected, as the multi-utilities contain a combination of electricity and natural gas operations. Interestingly, the multi-utility industry has little commodity risk exposure, suggesting either effective hedging strategies or economy of scope; diversified operations allow multiutilities to re-prioritise operations when faced with regulatory changes or commodity price fluctuations.
The following two sections outline tests concerning evolving risk premia through time. comparison with the more sophisticated approach presented in Section 3.5.

Time-varying Risk Premia: Annual Regressions
The established literature has reported substantial inter-temporal and inter-sectoral variability in the relationships between average returns and risk factors (Faff and Brailsford 1999;Sadorsky 2001;El-Sharif et al. 2005;Oberndorfer 2009;Fama and French 1997;Fama and French 1998;Fama and French 2012). To address time-varying risk loadings, the time series is separated into annual periods and the local AFFM, Equation (3), is applied annually. This provides results comparable with Tulloch, Diaz-Rainey, and Premachandra (2017a) and El-Sharif, Brown, and Burton (2005). Table 3 reports the annual regression results for the energy sector portfolio, estimated using Newey-West HAC standard errors. Regression coefficients through time and 95% confidence intervals are shown in Figure 3.
Overall, the results indicate evolving risk premia over the entire time series. Further, the intertemporal AFFM improves the goodness of fit, from an adjusted 2 of 72.77% in the unconditional regression of Table 2 to a mean adjusted 2 of 74.52% in Table 3.
The coefficients for size premia are negative and significant for all 18 years tested. The results show that the average returns in the energy sector behave like big energy utilities through time.
The relationship is not stable through time; there are years where the coefficient becomes more negative, suggesting that the returns of big utilities have greater influence on overall sector returns. In Table 3 Tulloch, Diaz-Rainey, and Premachandra 2017b).
Electronic copy available at: https://ssrn.com/abstract=3038712 For the most part, commodities explain very little in energy sector returns.  (2005); there is varying commodity risk exposure across time. The number of significant commodity coefficients increases in later years, possibly reflecting the uncertainty of rapidly changing commodity prices over the period.

Time-varying Risk Premia: Inductive Structural Break Point Tests
This section implements the Bai and Perron (1998; Table 4 present the stability diagnostic tests, while the results in Table 5 present the results of the break point regression. The break conditions of the two tests are also outlined in Table 4 and Table 5.  Table 5. The result shows that the estimated break points are identical to those in Table 4, identifying eight structural breaks in the time series. The Bai and Perron (1998; break point regression provides marked improvements to regression fits for the energy sector. The initial full-period global AFFM of produced an adjusted 2 of 68.79% (Table 2). The adjusted 2 of the inter-temporal global AFFM varied between 28.24% and 85.63%, with a mean of 69.64% (Table 3). When controlling for inductive structural breaks, the Bai and Perron (1998; break point regression increases the adjusted 2 of the AFFM to 80.42%, greater than the mean adjusted 2 values of 74.52% from annual regressions in Table 3 returns. Oil price risk shows increased significance immediately before the GFC, showing that returns in the energy sector were highly sensitive to oil price risk. Coal price risk has sporadic significance across time which continues to be negative (when significant). Natural gas price risk continues to be insignificant through time, again, indicating effective hedging strategies. The differences between these results and the deductive method of Table 3 are manifestations of Quandt's (1960) criticisms: unless the break points are known with certainty, the significance tests and estimated coefficients are likely to be biased.

Better Isolate Firm-specific Returns: Inductive Structural Break Point Tests
The fourth analytical foci of this paper address better isolating the firm-specific returns of sector-level returns. This approach can be useful in applications that require more precise estimates of expected stock return, including 1) portfolio selection, 2) evaluating portfolios performance, 3) estimates of cost of capital and 4) measuring abnormal return in event studies (Fama and French 1993). Specifically, Tulloch, Diaz-Rainey, and Premachandra (2017a) utilise the local AFFM to perform an event study analysis on the impact of liberalization and environmental policy on the financial returns of European energy utilities.
The standard approach includes extracting the residuals from a regression, for example, the unconditional CAPM, where the residuals are assumed to be the unsystematic, or firm-specific, component of returns (Chan, Chen, and Hsieh 1985;Fama and French 1993). However, any omitted variables from the model specification will bias the estimates of regression coefficients, while intercorrelated disturbances will bias standard errors; both result in a false positive.
Typically, the three-and four-factor models capture common returns across a variety of stocks and are better at isolating the firm-specific components of returns (Fama and French 1993). The assumption that these parameters are stable can result in spurious significant correlations between residuals and exogenous risk factors. Fama and French (1993) have also drawn inferences based on this fallacy, performing residual diagnostic tests with constant slopes; however, the authors acknowledge that the assumption on constant slopes may be a misconception but do not investigate the claim further. The following paragraph demonstrates the consequences of this erroneous assumption.
The residuals of the local AFFM model are extracted using three different methods, including (1) the assumption of constant parameters in the unconditional local AFFM regression of Table 2 similar to Fama and French (1993); (2) the time-varying parameters of the conditional annual local AFFM regressions in Table 3 similar to the El-Sharif, Brown, and Burton (2005), who did not perform additional residual diagnostic tests; and (3) time-varying parameters of the local AFFM sequential break point test in Table 5 based on the minimisation of the SSR from Bai and Perron (2003).
The cumulative and daily residuals through time are shown in Figure 4. The first observation is that the residuals of method (1), the unconditional constant parameter assumption, differ greatly from the other two, time-varying assumptionmethods (2) and (3). For method (1), the assumption of constant parameters results in the cumulative residuals for energy utilities reaching 200% by mid-2008. The first invalid inference would be that this represents a firm-specific component to returns, increasing from 2003 to the GFC, and decreasing thereafter. It could also be hastily concluded that this shift represents a structural break after the second packages of liberalisation (in 2003), expected to be a major regulatory event. Further, empirical tests will also identify significant breaks in returns which coincide with such an interpretation, in 2003 and 2008, as shown in Table 6.
Although the structural breaks are significant, the adjusted 2 shows that these breaks explain a small proportion of the variation in the residuals, the fraction of variance unexplained. In contrast to the significant structural breaks in the unconditional method (1), the time-varying methods (2) and (3) find no structural breaks. How can this result be reconciled? An explanation may be that the residuals of the unconditional approach in method (1), in fact, contain systematic risk factors beyond the firmspecific components of returns. The residuals reflect the changing relationship between the stock returns and the risk factors. This can be demonstrated in two ways. First, the time-varying methods, (2) and (3), both allow the risk factors to vary across time, capturing parameter shifts. Table 6 and Figure 4 show that the residuals in these two methods fall close to zero, suggesting few impacts from factors beyond those specified in the mean equation. Second, the expectation is that the residuals of the unconditional regression should have filtered out the impact of all risk premia, leaving an orthogonalised return series which represents only the firm-specific element of returns.
A second pass of the unconditional residuals in method (1), using either linear regressions or sequential break points model specifications, is expected to have no relationship with the risk factors, as they should have been filtered out in the first pass. However, this is not the case. To demonstrate, a second sequential break point test is employed on the unconditional residuals of method (1) against the market factor, local stock market risk factors, term premium and commodity risk factors; results are shown in Table 7. The first observation is the presence of significant relationships between the unconditional residuals and the risk factors. The adjusted 2 shows that the unconditional full-period regression is insufficient at removing the risk premia, explaining 27.95% of residual variation. The economic rationale is simple, a linear relationship is insufficient to capture time-varying common risk over long horizons; the polygonal curve-fitting approach of Bai and Perron (2003) is able to control for the non-linearity and partial breaks across time.
The dates of the sequential break points in Table 7 are identical to those in Table 4 and Table   5, giving confidence that the residuals are capturing a shift in the relationship with risk premia. The estimated coefficients in Table 7 represent the difference between the estimated coefficients of the unconditional local AFFM (Table 2) and the estimated coefficients of the local AFFM break point regression (Table 5). Put simply, the estimated coefficients in Table 7 represent the increasing or decreasing relationship between average returns and the risk premia during the partition. For example, the long-term market beta in Table 2 (also shown in Table 7) is 0.6306, while partition 2 of Table 7 estimates that the market beta experienced a statistically significant downwards shift of -0.3297, providing an overall market beta of 0.3009. Allowing for rounding, this value matches the estimated market beta of partition 2 in Table 5. In fact, the difference between each coefficient in Table 7 and the unconditional estimate equals the coefficient in Table 5.
This implies that the residuals of the unconditional regression still contain risk premia, failing to accurately represent the firm-specific component of returns over long horizons. These results show that asset pricing models must control for this changing relationship with premia over time to better isolate the firm-specific component of returns, necessary to calculate abnormal performance as a result of regulatory changes.
The overall results indicate that structural break points in parameters, previously ignored in the unconditional model assuming constant slopes, account for around 28% of the residuals which were previously assumed to be the firm-specific element of excess returns. These results demonstrate that the sequential partial break point approach has much greater ability to filter out systematic elements of returns.

Conclusion
This paper outlines a modelling approach for implementing sector asset pricing studies. This is achieved through the four analytical foci of the paper, namely: 1) by calculating stock market risk factors at the sector level, 2) by creating sub-group portfolios to explore within-sector heterogeneity; 3) by applying inductive structural break point tests to identify time-varying risk premia; and 4) by better isolating the firm-specific component of returns though a 'second pass' structural break regression on residuals. We do so by creating a comprehensive asset pricing model, the AFFM, which includes commodity and macroeconomic risk factors previously found to affect sector returns. We improve on the asset pricing model by calculating local risk factorsrisk factors which are more relevant to the sector of interest and better account for within-sector characteristics.
Electronic copy available at: https://ssrn.com/abstract=3038712 For analytical focus 1, the use of local stock market risk factors, specific to the energy utility sector, improves the performance of asset pricing models and can explain a greater proportion of average returns at the sector level. Our results show the asset pricing models which use local risk factors had greater explanatory power as regression fits were tight, resulting in higher 2 values, increasing adjusted 2 from 68.79% using the standard risk factors to 72.77% using the local risk factors.
Addressing analytical focus 2, we grouped the energy stocks into 12 sub-sector portfolios based on company characteristics. Our results show substantial heterogeneity across the 12 portfolios.
The spread in estimated coefficients show that the heterogeneous sensitivity to size, value and momentum premia are the largest determinants for the differences in expected returns across various energy portfolios. Interestingly, the multi-utility portfolios show one of the lowest cumulative abnormal returns across all portfoliospossibly indicating a lower risk-return relationship. This is consistent with economy of scopediversified operations allow multi-utilities to switch operations when faced with regulatory changes or fluctuations in commodity prices.
The third analytical focus examines how to account for time-varying risk premia using deductive conditional annual regressions and the inductive Bai and Perron (1998;  inductive structural break point test. This paper demonstrates that the common method of isolating the residuals of the unconditional regressions, such as the method implemented by Fama and French (1993), performs poorly at capturing the firm-specific components of returns. This is because the assumption of constant parameters forces the changing relationship between the sample returns and the risk factors into the residuals of a linear regression. We examine this proposition using the residuals of an unconditional local AFFM and applying a 'second pass' test on these residuals using the Bai and Perron (1998; structural break point regression. Our results, specific to this paper's energy sample and time period, show that 28% of the residuals variance is related to the changing relationship between sector returns and risk factors. We detect structural breaks in the residuals of an unconditional regression, which correlate with structural breaks in the relationship between sector returns and risk premia. We show that a better method of isolating the firm-specific component of returns is to account for multiple structural changes in model parameters using polygonal curves and minimising variance within groups.
Overall, this paper provides a template for conducting sector asset pricing studies which can be adapted to other sectors, since it suggests an approach of more accurately examining time-varying risk premia and isolating the firm-specific component of returns for any sector. This is important not just to investors but also regulators in regulated sectors who want to understand the impact of policies on sector risk-return dynamics and cost of capital. As such, this approach could be applied to sectors such as banking and telecommunications, which are both regulated and undergoing dramatic technologically driven change.

CAPM, FOUR-FACTOR AND LOCAL AUGMENTED ASSET PRICING MODELS
This table presents the Newey-West regression output for the 12 energy portfolios against eight risk factors, using four model specifications. The 12 value-weighted portfolios include the energy sector ( , ), high-BE/ME utilities ( , ), mid-BE/ME utilities ( , ), low-BE/ME utilities ( , ), up momentum utilities ( , ), neutral momentum utilities ( , ), down momentum utilities ( , ), electricity utilities ( , ), natural gas utilities ( , ), multi-utilities ( , ), small utilities ( , ) and big utilities ( , ). The eight risk factors include market premium ( , ), local size premium ( ), local value premium ( ), local momentum premium ( ), term premium ( , ), oil risk ( , ), coal risk ( , ) and gas risk ( , ). For model 5, global risk factors are used in place of local risk factors. A ****, ***, ** or * denotes significance at 0.1%, 1%, 5% or 10%, respectively.    3: INTER-TEMPORAL ANALYSIS OF SECTOR PORTFOLIO USING THE LOCAL AFFM  This table presents conditional annual local AFFM, estimated on a year-by-year basis using Newey-West HAC standard errors between 1996 and 2013. The value-weighted returns of the energy sector ( ) is used as the dependent variable. The nine risk factors include market premium ( ), local size premium ( ), local value premium ( ), local momentum premium ( ), term premium ( ), oil risk ( ), coal risk ( ), natural gas risk ( ) and carbon risk ( 2 ). The intercept and error term is denoted and , respectively. A ****, ***, ** or * denotes significance at 0.1%, 1%, 5% or 10%, respectively. The specification used is: Electronic copy available at: https://ssrn.com/abstract=3038712  (2). The results are estimated using sequential evaluation, a maximum of 18 breaks and a trimming percentage of 5%. Repartition are suspected break dates. * Significance at ≤ 10% ** Critical values from Bai and Perron (2003 Bai and Perron (1998; sequential multiple partial break point tests. The HAC coefficient covariance matrix automatically determines optimised lag structuring using AIC. Kernel bandwidth is automatically determined using Andrew's AR(1) method and uses quadratic-spectral kernels. The break specification is sequential, testing the null of ℓ versus the alternative of ℓ + 1 breaks. The information criterion is set to allow a maximum of 18 structural breaks, employs a trimming percentage of 5%, and significance at ≤ 10%. The value-weighted returns of the energy sector ( ) is used as the dependent variable. The eight risk factors include market premium ( , ), local size premium ( ), local value premium ( ), local momentum premium ( ) term premium ( , ), oil risk ( , ), coal risk ( , ) and gas risk ( , ). The intercept and error is denoted and , , respectively. A ****, ***, ** or * denotes significance at 0.1%, 1%, 5% or 10%, respectively. The specification used is:   Bai and Perron (1998; sequential multiple 'pure' break point tests. The HAC coefficient covariance matrix automatically determines optimised lag structuring using AIC. Kernel bandwidth is automatically determined using Andrew's AR(1) method and uses quadratic-spectral kernels. The break specification is sequential, testing the null of ℓ versus the alternative of ℓ + 1 breaks. The information criterion is set to allow a maximum of 18 structural breaks, employs a trimming percentage of 5%, and significance at ≤ 10%.   Bai and Perron (2003) sequential multiple partial break point tests on the residuals of the constant slope regression. The HAC coefficient covariance matrix automatically determines optimised lag structuring using AIC. Kernel bandwidth is automatically determined using Andrew's AR(1) method and uses quadratic-spectral kernels. The break specification is sequential, testing the null of ℓ versus the alternative of ℓ + 1 breaks. The information criterion is set to allow a maximum of 18 structural breaks, employs a trimming percentage of 5%, and significance at ≤ 10%. The residuals of the constant slope regression is used as the dependent variable. The eight risk factors include market premium ( , ), local size premium ( ), local value premium ( ), local momentum premium ( ), term premium ( , ), oil risk ( , ), coal risk ( , ) and gas risk ( , ). The intercept is denoted . A ****, ***, ** or * denotes significance at 0.1%, 1%, 5% or 10%, respectively. The specification used is: = + , + + ℎ + + , + , + , + , + . Where represents the true firm-specific component of returns.