Open access peer-reviewed chapter

# An Application of Wavelets to Finance: The Three-Factor Fama/French Model

Written By

Bruce D. McNevin and Joan Nix

Submitted: October 8th, 2017 Reviewed: January 18th, 2018 Published: October 3rd, 2018

DOI: 10.5772/intechopen.74165

From the Edited Volume

## Wavelet Theory and Its Applications

Chapter metrics overview

View Full Metrics

## Abstract

We use multi-scale analysis and a rolling 250-day window to estimate a widely used standard for empirical asset pricing. The asset pricing model employed is the Fama-French three-factor model. The model is estimated using stock returns for 49 industry stocks of US industry portfolios for the period from July 1969 to September 2017. The rolling window estimation approach allows us to capture the behavior of an investor who periodically reallocates his portfolio. Employing periodic estimates of expected return, we implement a set of long/short investment strategies based on the standard Fama-French three-factor model, and scale versions of the model. We find that during recessions, the higher scale long/short strategies tend to outperform the standard approach. Our results suggest distinct risk dynamics at specific horizons during recessions. We conclude that the information content of the economic phenomena that generate the three-factor model does not follow strict periodicity during recessions, making the wavelet approach more suitable for portfolio managers who must be prepared to rebalance portfolios during official downturns.

### Keywords

• wavelets
• portfolio returns
• investment horizon

## 1. Introduction

The Holy Grail of finance is an empirical asset pricing model that explains stock returns. Most models fall under the risk/return umbrella where risk is positively related to return. There are two basic models in empirical asset pricing, the standard Capital Asset Pricing Model, CAPM [15, 17, 21] and the Fama/French three-factor model, FF3 [5]. The basic idea behind the CAPM is that market movements matter a lot for capturing the relationship between risk and return. The systematic risk measure, beta, is an estimate of the sensitivity of a security or portfolio’s returns to market movements. In the risk/return world, the CAPM is considered a one-factor model in that a single factor, the market return, does all the heavy lifting. The model specification is as follows:

ritrft=αi+βirmtrft+etE1

where rit = return of firm i at time t, rmt = market return at time t, and rft = risk free rate at time t. The slope term, βit, estimates systematic risk. The intercept, αi, measures abnormal returns, or returns not explained by market exposure of the security or portfolio. In the context of the CAPM, αi is expected to be zero since only non-diversifiable, also referred to as systematic or market risk, represents the risk that matters for explaining returns.

While the CAPM remains a cornerstone of financial theory, numerous empirical studies have called into question the ability of the CAPM to explain the cross-section of expected stock returns (see for instance, [3]). Several studies have used wavelets to examine the CAPM across scale. Gencay [7] first proposed the use of wavelets to estimate systematic risk in the Capital Asset Pricing Model. They estimate the beta of each stock annually for 6 wavelet scales using daily returns for the period January 1973 to November 2000 for stocks that were in the S&P 500. They find a positive relationship between portfolio returns and beta. Gencay et al. [8] extend their 2003 study by including stocks from the Germany and UK. They find that scale matters in other markets in that the relationship between portfolio returns and beta becomes stronger at high scales. Fernandez [6] applies wavelet analysis to a model of the international CAPM using a data set that consists of daily aggregate equity returns for seven emerging markets for the period 1990–2004.1 The ICAPM2 was estimated at 6 scales (2–128 day dynamics). Fernandez finds that market sensitivities are generally greatest at the higher scales of 5 and 6. In addition, the R2 peaked at scales 5 and 6. She concludes that the ICAPM does its best at capturing the relationship between risk and return at the medium scale or long-term scale that for their data set is 32–128 days. An important takeaway from research employing wavelet measures of beta is that when the environment is distinguished by slowly changing features, or low frequency events the CAPMs’ applicability in terms of providing a measure of systematic risk improves when using wavelets. This is consistent with the findings of Rua and Nunes [20] that employs wavelet methodology and provides evidence that market risk varies across time and over frequencies.3

The adage the proof of the pudding is in the eating is of particular relevance for empirical asset pricing models. Practitioners want to know if they employ a specific empirical asset pricing model will their investors benefit? The fierce competition to develop a winning model continues among various market players, especially hedge funds [2]. The prescription to basically accept that markets are efficient and form a portfolio that passively tracks the market has contributed to the growth of index investing, but has not slowed the search for a better model. The idea of basically finding other factors besides the market that explain equity returns has generated many different versions of factor models. One that has gained widespread acceptance is the Fama and French three-factor model (FF3). The general consensus is that the FF3 has greater explanatory power than the CAPM. The Fama-French model adds to the explanatory power of the standard CAPM by including two additional factors, firm size and the book-to-market ratio. Both factors were found in previous research to matter for explaining equity returns. That small firms outperform large cap firms is found in Banz [1], while Barr Rosenberg, Kenneth Reid, and Ronald Lanstein [19] find a positive relationship between average stock returns and book-to-market ratio. Low B/M firms are considered “value stocks” while high B/M are “growth stocks.” There is strong consensus around the idea that smaller cap firms are riskier and therefore, generating greater returns beyond what would be expected from simple market beta exposure is a widely accepted explanation for the size factor. There is less agreement for an explanation of the value premium, but one is rooted in behavior where basically relatively cheap stocks outperform relatively expensive stocks because optimism and pessimism persist among investors. Investors bid up growth stocks leading to future under performance, and keep down value stocks leading to future over-performance. Both size and B/M factors are added to their model as factors that account for returns, along with the market factor as found in the CAPM. The FF3 model is specified as follows:

ritrft=αi+βirmtrftbeta2iSMBt+β3iHMLt+etE2

where SMBt and HMLt are the size and book-to-market factors, respectively. The book-to-market ratio is intended to capture the difference between value and growth stocks in the sense that the book-to-market ratio is high for value stocks and low for growth stocks.4

Several studies have examined the Fama-French 3-factor model at the scale level. Kim and In [10–14] apply wavelets to the Fama-French 3-factor model using monthly data from 1964 to 2004 for 12 industry portfolios. They find that the market variable plays an important role in explaining stock returns across all scales. In addition, they find that the estimated coefficients for the SMB and the HML are significant in specific time scales, depending on the industry. Trimtech et al. [22] apply wavelet analysis to the Fama-French model to study monthly returns for the French stock market for the period 1985. They find that the r-square of the medium and high scale versions of the Fama-French model exceed that of the standard model. They also find that the risk sensitivity of the factors depends on the time scale with the magnitude and sign of the size and book-to-market factors varying across scale.

We use multi-scale analysis and a rolling 250-day window to estimate the Fama-French 3-factor model of stock returns for 49 industry stocks of US industry portfolios. The data set, which consists of daily observations, covers the period from July 1, 1969 to September 29, 2017. We find through risk-adjusting the portfolios using the FF3 model that there are distinct risk dynamics during recessions. The rolling window estimation approach allows us to capture the behavior of an investor who periodically reallocates his portfolio. Using periodic estimates of expected return we implement a set of out-of-sample long/short investment strategies based on the standard Fama-French model, and also the scale versions of the model. We find that for the sample as a whole the strategy based on the standard model outperforms each of the scale based strategies. In other words, frequency-based information does not appear to matter for portfolio performance when spanning the entire time period. However, during the majority of recessions, the higher scale long/short strategies tend to outperform the standard approach. The frequency content of information does appear to matter during recessions. We conclude that most recessions reflect a time-varying market regime where scale dynamics matter for portfolio performance. In terms of practioners the results suggest that an avenue for potential improvement in portfolio performance is found by taking scale into consideration when faced with potential recessionary periods.

The remainder of this chapter is organized as follows: Section 2 presents the data and basic statistics. Section 3 describes the methodology. Section 4 presents the empirical findings, and Section 5 follows with our concluding comments.

## 2. Data discussion

Our analysis uses daily equity returns for 49 value-weighted industry portfolios for the period July 1, 1967 to September 29, 2017. The portfolios, which are made available by Kenneth French at his website,5 are defined by assigning each NYSE, AMEX, and NASDAQ stock to an industry at the end of June in year t, using Compustat 4 digit SIC codes for the fiscal year ending in calendar year t−1. The industry definitions, along with basic statistics for daily returns, are provided in Table 1. The returns, which are shown in excess of the risk free rate, range from a low of 0.002% for Real Estate to a high of 0.0522% for Tobacco. The sign of the skewness varies across industries, but the returns for all industries are leptokurtotic.

SectorNameIndustryMeanStd.DevSkewnessKurtosis
ChemicalsChemsChemicals0.03101.2503−0.1428.77
Consumer DurablesHshldConsumer goods0.02271.0970−0.2369.76
Consumer Non-DurablesAgricAgriculture0.03051.41280.39014.57
Consumer Non-DurablesBeerBeer & liquor0.03591.1521−0.40114.72
Consumer Non-DurablesBooksPrinting and publishing0.02191.2104−0.28410.71
Consumer Non-DurablesClthsApparel0.02661.2706−0.0576.84
Consumer Non-DurablesFoodFood products0.03340.9178−0.0449.93
Consumer Non-DurablesSmokeTobacco products0.05231.4031−0.3667.11
Consumer Non-DurablesSodaCand & soda0.03501.4357−0.28410.67
Consumer Non-DurablesToysRecreation0.01631.4800−0.03518.54
Consumer Non-DurablesTxtlsTextiles0.02681.3612−0.84322.88
EnergyCoalCoal0.03182.4043−0.1878.55
EnergyMinesNon-metallic and metal0.02741.6273−0.3559.47
EnergyOilPetroleum and natural gas0.03061.3598−0.1263.94
HealthDrugsPharmaceutical products0.03411.15450.20510.03
HealthHlthHealthcare0.02431.52700.47112.22
HealthMedEqMedical equipment0.03091.18570.1177.20
ManufacturingAeroAircraft0.03681.3506−0.1979.90
ManufacturingAutosAutomobiles and trucks0.02131.4643−0.2696.62
ManufacturingBoxesShipping containers0.02941.2786−0.31910.36
ManufacturingElcEqElectrical equipment0.03551.3878−0.38010.49
ManufacturingFabPrFabricated products0.01501.5073−0.4419.15
ManufacturingGunsDefense0.04241.37980.24616.64
ManufacturingLabEqMeasuring and control equip.0.02901.4337−0.30710.14
ManufacturingMachMachinery0.02721.3123−0.1227.25
ManufacturingrubbrRubber and plastic products0.02791.1525−0.1316.20
ManufacturingSteelSteel works, etc.0.01651.6334−0.2369.17
MoneyBanksBanking0.02951.4384−0.1846.51
MoneyInsurInsurance0.03131.1682−0.48411.24
MoneyRlEstReal estate0.00221.5172−0.3558.30
OtherBldMtConstruction materials0.02821.2248−0.3067.41
OtherCnstrConstruction0.02461.5836−0.1756.79
OtherFunEntertainment0.04291.66880.34220.74
OtherGoldPrecious metals0.02442.3694−0.01816.42
OtherMealsRestaurants, hotels, motels0.03011.26840.29916.98
OtherOtherAlmost nothing0.00301.42950.22615.55
OtherTransTransportation0.02731.2429−0.16111.68
ShopsPerSvPersonal services0.00891.3192−0.09213.46
ShopsRtailRetail0.03051.1679−0.4386.14
ShopsWhlslWholesale0.02451.0612−0.5289.46
TelecommunicationsTelcmCommunication0.02661.1191−0.16313.71
UtilitiesUtilUtilities0.02390.87430.01221.07

### Table 1.

Daily return statistics (%), July 1, 1967 to September 29, 2017.

The period of analysis cover five recessions, which are listed in Table 2. Our analysis of the performance of the long/short portfolios across scale focuses on these five recessions.

PeriodDuration (mos)
Nov 1973–Mar 197516
Jan–July 19806
July 1981–Nov 198216
July 1990–Mar 19918
Mar 2001–Nov 20018
Dec 2007–June 200918

### Table 2.

Recessions and duration in data sample.

Excess market returns (Mkt), the risk free rate (RF), and the 2 Fama-French factors (SMB and HML) are also from Kenneth French’s website. Excess market returns include all NYSE, AMEX, and NASDAQ firms. The risk free rate is the 1-month Treasury bill rate. The two Fama-French factors are constructed using 6 value-weighted portfolios formed on size and book-to-market. The size factor, SMB (small minus big) is the average return on the three small portfolios minus the average return on the three big portfolios. Similarly, HML (high minus low) is the average return on the three value portfolios minus the average return on the three growth portfolios. Table 3 contains summary statistics for Mkt, RF, SMB, and HML. The average return for the HML portfolio exceeds that of the SMB portfolio. The HMB portfolio has a small negative skew while Mkt and SMB each have a positive skew. The kurtosis for the SMB portfolio is relatively large.

MKtSMBHML
Mean0.02530.00320.0171
Std. Dev.1.02480.54350.5217
Skewness−0.5049−1.06050.3507
Kurtosis14.811623.23729.9377

### Table 3.

Summary statistics for model factors, daily data, July 1, 1967–September 29, 2017.

Figures 1 and 2 contain the continuous wavelet power plots and time series plots of returns for Mkt, SMB, and HMB, respectively. For all three series the power tends to be highest for periods less than 256 days. Of the three series, HMB has the highest volatility of returns, and it tends to cluster around the recessionary periods. This is particularly true for the last two recessions. The SMB series has the lowest volatility, however, its power also tends to be highest during recessions.

## 3. Methodology

Our analysis of industry returns uses the Maximal Overlap Discrete Wavelet Transform (MODWT). The MODWT is calculated using a pyramid algorithm. Given a data series xt, a high pass wavelet filter h1˜, and a low pass scaling filter g1˜ are applied to obtain wavelet coefficients w1˜, and scaling coefficients v1˜. In the second step of the pyramid, the original data series xt is replaced by v1˜ which is passed a high pass filter h2˜ and a low pass filter g2˜ to obtain wavelet and scaling coefficients, w2˜, and v2˜, respectively. This procedure is repeated up to J times where J = log2(N). An important feature of the MODWT is that it can be applied to any sample size, while the Discrete Wavelet Transform (DWT) can only be applied to series of size 2J.6

We apply MODWT to each portfolio of industry returns, as well as, the market returns (MKT), the size returns (SMB), and the book-to-market returns. For a filter we choose the Daubechies orthonormal compactly supported wavelet of length L = 8 [4], least asymmetric family. We selected J = 6, common practice in wavelet applications to empirical asset pricing models for providing a good balance in the time and frequency localization. The investment horizons we evaluate cover 2–4 days (J = 1) to 64–128 days (J = 6).

### 3.1. Selecting a filter

In this section, we briefly discuss the process involved in selecting a filter. While our empirical analysis is primarily focused on results using a Daubechies Least Asymmetric filter of length L = 8, LA(8), we also provide results for two other filters to reflect the sensitivity of our results to the filter choice. These two alternative filters are the Daubechies extremal phase filter of length L = 4, DB(4), and the Coiflet filter of length L = 6, C(6).

Percival and Walden [18] point out that in selecting a filter there are two primary considerations, (1) if the filter length is too short it may introduce undesirable anomalies into the results; (2) if the filter is too long more coefficients will be affected by the boundary condition, and there will also be a decrease in the localization of the coefficients. They suggest using the smallest possible filter length that gives reasonable results. They also suggest that if one requires the filter coefficients to be aligned in time, as we do in or analysis, then the LA(8) is generally a good choice. It is not surprising that the LA(8) filter is a very common filter choice in research that applies wavelet methodology to finance.

Figure 3 compares the LA(8) wavelet filter with the two alternative filters used in our analysis. The filter lengths range from 4 to 8. The DB(4) filter has two vanishing moments; the Coiflet(6) has two vanishing moments and is nearly symmetric; the LA(8) has four vanishing moments. The greater the number of vanishing moments the smoother is the scale function.

Since our analysis employs the MODWT, we expect the results to be less sensitive to the filter choice than if we had used a DWT. As discussed in [18] MODWT details and smooths can be generated by averaging circularly shifted DWT details and smooths generated from circularly shifted time series. The averaging smooths out some of the choppiness that is found in DWT MRAs.7

### 3.2. Model specification

The specification of the Fama-French model that we estimated is as follows:

ritλjrftλj=aiλj+βiλjRMtλjRFtλj+β2iλjSMBtλj+β3iλjHMLtλj+eitλjE3

where λ=2j1, for j = 1, …,6. ritλjrftλj is the excess return for industry portfolio i and time t, and scale j. RMtλj, RMtλj, SMBtλj, and HMLtλj are the Fama-French factor for scale, j.

After we disaggregate the series to scale we use a rolling 250-day window to estimate the standard model, and each of the six scale level models. Each time we estimate the models we calculate the expected return for each industry as of the last day of the estimation period. We then rank the expected returns for that estimation period and assign a decile. The long-short strategy that we employ consists of going long (buying) the top decile, and going short (selling) the bottom decile. This position is held for 20 days. At the end of the 20 days period we re-estimate the models using the previous 250 days and repeat the investment selection process. Since there are 49 industry portfolios, this means that every 20 days we create a portfolio that is long 5 industries and short 5 industries. We calculate the out-of-sample cumulative returns for each 20-day period. We roll this process forward for the entire sample period.

## 4. Empirical findings

Our discussion of the empirical findings consists of four parts. We begin with a comparison of the parameters for the standard model parameters and the 6 scale models for the LA(8) filter. We discuss both sector averages, and industry results. Next, we examine parameter estimates for the alternative filters, DB(4) and C(6). We then discuss the returns for the long/short strategy at each scale over the entire sample period. Finally, we turn our focus to the performance of the strategies during periods of recession.

### 4.1. Parameter estimates

#### 4.1.1. LA(8) filter

Table 4 contains sector level averages of the industry ‘beta’ parameter estimates. The difference between the standard model and the scale models for the industries tends to be modest. This is generally consistent with studies that have used monthly data to evaluate sector returns across scale. For instance using the CAPM, McNevin and Nix [16] found only small differences between the standard beta and wavelet betas for scales 1 and 2.

SectorStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals1.0831.0491.0861.1021.1361.1761.084
Consumer Durables0.8490.8740.8310.8220.7640.7690.857
Consumer Non-Durables0.8890.8870.8910.8900.8970.8790.907
Energy1.1081.0921.1401.1411.1021.1891.018
Health0.9550.9630.9860.9600.9450.9480.866
Manufacturing1.0611.0511.0461.0831.0921.1101.070
Money1.0911.0621.0861.1231.1331.1091.214
Other1.0150.9961.0221.0511.0431.0330.982
Shops1.0141.0201.0151.0121.0311.0380.987
Telecommunications0.8880.9410.8810.8670.8300.8490.805
Utilities0.7090.7070.7130.7290.7390.7080.728

### Table 4.

Average Beta parameter by sector—LA(8).

Table 15 (in Appendix) contains the industry level parameter estimates of the market variable, or the ‘betas’. These parameters are averages of the rolling window estimates. There were a total of 597 rolling window regressions. On average, all of the parameter estimates in Table 15 are significant at the 95% level of confidence. Table 16 contains the corresponding t-statistics. There is no definitive pattern to the parameters across scale, though they tend to increase with scale.

Table 5 contains average sector parameters for the size variables. The range of parameters for the Business Equipment sector is the greatest, ranging from 0.092 for scale 1 to 0.463 for scale 6. Most of the other sectors do not exhibit a strong pattern across scale. The parameter estimates for utilities change sign across scale. In this case the sector and industry parameters are the same. An examination of Table 18 indicates that the standard model size parameter is insignificant for the utilities, but the parameters for scales 4–6 are all negative and significant. As shown in Table 17, the size parameter at the industry level can vary quite a bit across scale and in comparison to the standard model indicating that in some industries investors require a premium for investing in small firm stocks over longer investment horizons. Some examples include Chips, Software, Mines, Steel, Gold, and Lab. equipment.

SectorStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals0.0660.0410.0900.1100.0880.0290.041
Consumer Durables−0.270−0.289−0.290−0.275−0.175−0.132−0.207
Consumer Non-Durables0.1530.1660.1470.1340.1350.1880.146
Energy0.2470.2310.2480.2910.3480.2750.303
Health0.1580.1860.1640.1390.1140.1200.179
Manufacturing0.2600.2490.2420.2970.2810.2760.291
Money0.2860.3040.2910.2490.2160.1900.244
Other0.3620.3420.3750.3820.3660.3540.408
Shops0.3530.3690.3580.3180.2680.3200.384
Telecommunications−0.196−0.168−0.223−0.207−0.211−0.249−0.091
Utilities−0.0310.029−0.018−0.031−0.147−0.231−0.337

### Table 5.

Average size parameter by sector—LA(8).

Table 6 contains the average sector parameter estimates for the book-to-market factor. Two sectors with notable differences across scale are Chemicals and Energy. The Chemical sector only contains a single industry. Table 20 shows the t-statistics for the HML parameter at the industry level. On average, for the standard model the HML parameter is not statistically significant. However, it is positive and significant at scales 3–6. Table 19 contains the industry level parameters for the HML risk factor. As is the case with SMB, the importance of the HML factor across scale varies widely by industry. Notable difference across scale can be seen in Coal, Lab. Equipment, and Construction.

SectorStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals0.1930.1450.1890.2280.2700.2420.347
Consumer Durables−0.231−0.235−0.231−0.222−0.260−0.388−0.227
Consumer Non-Durables−0.021−0.013−0.016−0.037−0.042−0.0690.028
Energy0.4440.3550.4890.4950.5540.5780.653
Health−0.342−0.268−0.328−0.395−0.394−0.411−0.489
Manufacturing0.1880.2020.1760.1970.1890.1030.152
Money0.3920.3800.3800.3600.3670.3940.442
Other0.0770.0700.0770.0950.0740.0710.155
Shops−0.0140.025−0.003−0.039−0.035−0.072−0.114
Telecommunications0.2530.3050.2750.2260.1960.2740.097
Utilities0.4180.3720.4220.4350.4670.5130.395

### Table 6.

Average book-to-market parameter by sector—LA(8).

#### 4.1.2. Alternative filter parameter estimates: DB(4), C(6) filters

In this section, we provide sector averages of parameter estimates for the Fama-French model based on two alternative filters.8 Tables 7 and 8 contain the average sector betas for the DB(4) and C(6) filters, respectively. The sector level averages for the two alternative filters are quite similar. What is important for our analysis is that they are similar to the results for the LA(8) filter (Table 4). Tables 9 and 10 contain the sector parameter estimates for the firm size variable for the DB(4) and C(6) filters, respectively. These parameter estimates are also similar across filters. Tables 11 and 12 show the parameters for the book-to-market variable for the alternative filters. In summary, there is very little difference in parameter estimates across the different filters.

SectorScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals1.0511.0861.1031.1311.1641.109
Consumer Durables0.8690.8350.8240.7740.7810.849
Consumer Non-Durables0.8870.8920.8890.8960.8820.905
Energy1.0951.1331.1391.1081.1611.053
Health0.9640.9830.9610.9460.9490.879
Manufacturing1.0501.0501.0791.0911.1071.085
Money1.0631.0861.1191.1301.1161.201
Other0.9961.0211.0481.0431.0261.000
Shops1.0181.0161.0121.0271.0341.002
Telecommunications0.9370.8830.8630.8350.8480.814
Utilities0.7080.7130.7270.7350.7010.714

### Table 7.

Average Beta parameter by sector—DB(4).

SectorScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals1.0511.0861.1031.1321.1681.105
Consumer Durables0.8700.8350.8250.7730.7800.848
Consumer Non-Durables0.8870.8910.8890.8960.8820.907
Energy1.0951.1331.1411.1091.1621.043
Health0.9640.9830.9610.9470.9480.878
Manufacturing1.0501.0491.0791.0911.1081.082
Money1.0631.0861.1191.1311.1171.202
Other0.9961.0211.0481.0431.0281.000
Shops1.0181.0161.0121.0281.0351.002
Telecommunications0.9370.8830.8640.8350.8470.814
Utilities0.7080.7130.7270.7350.7010.716

### Table 8.

Average Beta parameter by sector—C(6).

SectorScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals0.0430.0830.1080.0890.0260.042
Consumer Durables−0.290−0.288−0.274−0.187−0.143−0.211
Consumer Non-Durables0.1650.1480.1340.1350.1750.131
Energy0.2340.2450.2940.3340.2860.309
Health0.1850.1640.1400.1210.1220.147
Manufacturing0.2490.2440.2920.2840.2750.292
Money0.3030.2910.2530.2210.1900.232
Other0.3430.3700.3800.3650.3600.395
Shops0.3690.3550.3180.2780.3180.371
Telecommunications−0.172−0.215−0.214−0.212−0.221−0.101
Utilities0.027−0.015−0.039−0.138−0.215−0.315

### Table 9.

Average size parameter by sector—DB(4).

SectorScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals0.0430.0830.1080.0890.0280.043
Consumer Durables−0.290−0.287−0.274−0.186−0.142−0.209
Consumer Non-Durables0.1650.1480.1360.1360.1780.135
Energy0.2330.2450.2920.3320.2790.305
Health0.1850.1640.1400.1200.1250.156
Manufacturing0.2490.2440.2910.2840.2740.296
Money0.3030.2910.2550.2220.1880.231
Other0.3430.3710.3810.3660.3570.397
Shops0.3690.3550.3190.2800.3190.375
Telecommunications−0.172−0.215−0.213−0.213−0.223−0.105
Utilities0.027−0.015−0.037−0.139−0.217−0.319

### Table 10.

Average size parameter by sector—C(6).

SectorScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals0.1470.1900.2270.2600.2520.348
Consumer Durables−0.236−0.226−0.221−0.261−0.356−0.225
Consumer Non-Durables−0.014−0.017−0.036−0.042−0.0610.022
Energy0.3660.4750.5000.5520.5700.624
Health−0.271−0.327−0.395−0.388−0.419−0.465
Manufacturing0.2010.1810.1950.1840.1170.160
Money0.3810.3820.3640.3750.3930.447
Other0.0700.0780.0930.0750.0760.146
Shops0.020−0.002−0.036−0.038−0.070−0.085
Telecommunications0.3020.2730.2210.2060.2520.105
Utilities0.3760.4200.4350.4680.5000.385

### Table 11.

Average book-to-market parameter by sector—DB(4).

SectorScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Chemicals0.1470.1900.2270.2610.2560.349
Consumer Durables−0.236−0.227−0.220−0.262−0.358−0.224
Consumer Non-Durables−0.014−0.017−0.036−0.042−0.0610.021
Energy0.3650.4760.4980.5530.5730.636
Health−0.271−0.328−0.392−0.389−0.419−0.464
Manufacturing0.2010.1800.1950.1830.1160.160
Money0.3810.3810.3630.3730.3910.442
Other0.0700.0780.0930.0750.0760.148
Shops0.021−0.002−0.037−0.040−0.072−0.090
Telecommunications0.3020.2730.2200.2070.2580.107
Utilities0.3760.4200.4350.4690.5020.391

### Table 12.

Average book-to-market parameter by sector—DB(4).

Our comparison of paramater estimates across filters provides support that our parameter estimates based on the MODWT are not over sensitive to the choice of a filter. The remainder of the chapter focuses on the results for the LA(8) filter—a filter that is widely used in finance research employing wavelet methodology.

### 4.2. Long-short strategy

In this section, we review the results of the long/short strategies applied over time. We begin by examining the average statistics for the out-of-sample results for both the standard Fama-French model and each of the scales. Table 13 presents a summary of the results.

StandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Mean2.47−0.600.40−0.541.71−0.700.85
Std. dev29.7528.2828.5027.2426.9225.8123.45
Skewness−0.340.18−0.09−0.07−0.09−0.29−0.20
Kurtosis2.481.601.731.682.501.620.90
Minimum−166.32−112.08−110.44−117.63−109.45−127.75−97.01
Maximum106.20115.68105.22100.11125.5487.2174.49
Median3.82−0.351.21−0.03−0.070.351.08
Sharpe Ratio0.08−0.020.01−0.020.06−0.030.04

### Table 13.

Average 20-day cumulative returns for long-short strategy—LA(8).

On average the cumulative 20-day return for the standard model (2.47%) exceeds all of the scale models. The scale 4 model has the second highest average cumulative returns (1.71%). The standard deviations are quite similar for all 7 models. The minimum and maximum cumulative returns are both quite high for all 7 models. This reflects the fact that there are only 10 positions in the out-of-sample portfolio at any point in time. It may also reflect the fact that the positions in the portfolio have equal weights (in absolute value). Finally, the Sharpe ratio for each of the models, even the standard model, is close to zero.

### 4.3. Strategy performance during economic recessions

While the scale level model does not seem to improve the long/short strategy overall, an examination of the returns during recessions tells a different story. As shown in Table 14 and Figures 46 for four of six recessions the returns at scale level exceed those using the standard model. In particular, the deep recession of the 1970s, as well as, the more recent financial crisis, illustrates how scale effects matter for designing portfolios that maximize returns (Figures 46 and Table 14).

RecessionBaseScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
Nov 1973–Mar 19750.781.353.941.280.225.970.10
Jan–July 19802.01.111.000.840.961.391.39
July 1981–Nov 19821.530.070.550.090.240.100.54
July 1990–Mar 19911.170.811.110.840.462.880.84
Mar 2001–Nov 20011.990.3012.340.050.080.035.35
Dec 2007–June 20090.000.010.00−0.410.110.002.35

### Table 14.

Cumulative out-of-sample returns during recessions—LA(8).

## 5. Conclusion

The focus of this chapter is on whether adding wavelet methodology to the FF3 model is really “worth it.” We attempt to show why it makes sense to add this methodology to the empirical asset pricing toolkit, and ultimately why practitioners should also consider including wavelet methodology in the mix of empirical asset pricing techniques used to provide advice and select portfolios for clients. The most fundamental reason for answering in the affirmative regarding whether wavelet methodology should have a seat at the table of empirical asset pricing models is that when an identified risk “signal” shows different behavior at different time periods, wavelet analysis, capable of decomposing data into several time scales, allows the researcher an opportunity to investigate the behavior of the risk factor/signal over various time scales. The exploration is richer because it allows windows to vary. Of course, allowing for risk measures that vary over time and across frequencies is not the same as finding that it will always matter for the results when compared to a standard approach devoid of such possibilities. Consistent with other research employing scale versions of the FF3 model, we find industry-specific effects on size and HML factors that are absent using the standard model. The large-scale versus fine-scale information distinction that the scale version of the FF3 model is capable of capturing is found significant for portfolio performance during the majority of recessions included in our data. Finding that the wavelet-based version of the FF3 model produces better portfolio outcomes is of importance to practioners, as well as, researchers. Our main conclusion based on the inter-temporal behavior of financial characteristics estimated with the FF3 model is that risk measures that vary over time and across frequencies are needed to capture the risk dynamics associated with most downturns. The importance of scale effects during periods defined as recessions leads us to conclude that the distinct risk dynamics during recessions are better captured with a methodology that allows for scale effects, providing yet another reason why wavelet methodology is a worthwhile tool that belongs in the methodological toolbox of practitioners in finance.

See Tables 1520.

SectorIndustryStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
ChemicalsChems1.0831.0491.0861.1021.1361.1761.084
Consumer DurablesHshld0.8490.8740.8310.8220.7640.7690.857
Consumer Non-DurablesAgric0.8630.8940.8440.8510.8940.8500.826
Consumer Non-DurablesBeer0.7490.7490.7620.7480.6890.7340.827
Consumer Non-DurablesBooks0.9130.8720.9160.9290.9321.0201.071
Consumer Non-DurablesClths1.0411.0231.0411.0801.0641.0230.982
Consumer Non-DurablesFood0.7320.7590.7410.7240.7010.7220.718
Consumer Non-DurablesSmoke0.8150.8140.8180.8280.8590.7560.817
Consumer Non-DurablesSoda0.8100.8460.7980.7410.7750.7420.892
Consumer Non-DurablesToys1.0721.0501.0661.0631.1381.0531.017
Consumer Non-DurablesTxtls1.0110.9771.0311.0471.0181.0121.011
EnergyCoal1.2111.2121.2661.2381.1271.2691.091
EnergyMines1.1081.0641.1381.1721.1921.2291.060
EnergyOil1.0041.0011.0161.0120.9881.0710.902
HealthDrugs0.8410.8500.8740.8370.8150.8670.793
HealthHlth1.0861.0901.1121.0931.1081.0731.030
HealthMedEq0.9380.9490.9720.9490.9110.9030.776
ManufacturingAero1.1281.1201.1371.1591.1501.1461.129
ManufacturingAutos1.2161.2081.1991.2201.2691.1931.151
ManufacturingBoxes0.9620.9630.9601.0050.9601.0000.986
ManufacturingElcEq1.0521.0461.0241.0781.0631.0911.030
ManufacturingFabPr1.0431.0451.0141.0731.1451.0591.080
ManufacturingGuns0.8680.8830.8160.8480.8630.9000.858
ManufacturingLabEq1.1131.0961.1071.1051.1261.1251.108
ManufacturingMach1.1461.1161.1451.1701.1891.2171.094
ManufacturingPaper0.9830.9710.9820.9921.0211.0591.100
ManufacturingRubbr0.9380.9270.9200.9670.9580.9700.979
ManufacturingShips0.9730.9490.9321.0141.0181.1671.045
ManufacturingSteel1.3091.2841.3161.3621.3481.3921.275
MoneyBanks1.1491.1021.1571.1771.1941.1631.269
MoneyFin1.1791.1351.1741.2271.2421.1871.337
MoneyInsur1.0150.9901.0241.0361.0611.0571.128
MoneyRlEst1.0211.0210.9881.0501.0351.0281.122
OtherBldMt1.0561.0241.0661.1001.0881.0571.085
OtherBusSv1.0281.0211.0331.0511.0641.0751.030
OtherCnstr1.2771.2391.3131.3351.3431.3241.249
OtherFun1.1551.1621.1821.1411.1311.1611.206
OtherGold0.4180.3370.3810.5600.5510.5080.303
OtherMeals1.0061.0111.0020.9901.0100.9670.926
OtherOther1.0301.0221.0451.0800.9791.0191.038
OtherTrans1.1491.1511.1511.1561.1771.1551.016
ShopsPerSv1.0451.0571.0391.0291.0731.1091.080
ShopsRtail1.0151.0161.0381.0081.0220.9690.944
ShopsWhlsl0.9820.9880.9681.0000.9981.0370.936
TelecommunicationsTelcm0.8880.9410.8810.8670.8300.8490.805
UtilitiesUtil0.7090.7070.7130.7290.7390.7080.728

### Table 15.

Average betas by industry.

SectorIndustryStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
ChemicalsChems21.92519.73821.79923.71325.05728.15127.495
Consumer DurablesHshld17.38016.02816.49818.20219.85222.05627.869
Consumer Non-DurablesAgric8.8218.0998.4909.47510.37711.38614.378
Consumer Non-DurablesBeer10.9329.91710.97511.54511.50813.50717.300
Consumer Non-DurablesBooks16.10313.93715.86518.20019.56322.65527.925
Consumer Non-DurablesClths16.41114.66316.09718.23719.71521.16321.696
Consumer Non-DurablesFood16.34215.23616.66218.01518.75019.28718.581
Consumer Non-DurablesSmoke9.3238.5509.0249.90611.15611.71512.592
Consumer Non-DurablesSoda8.7148.0108.3479.16110.21210.90814.548
Consumer Non-DurablesToys12.40810.77912.24813.78516.07516.38116.935
Consumer Non-DurablesTxtls14.76212.81814.96516.36417.84719.03519.842
EnergyCoal8.1147.3728.1928.7958.88410.58310.666
EnergyMines12.20810.83612.52213.96513.91015.22315.581
EnergyOil15.78915.53815.54116.15616.02118.74716.642
HealthDrugs16.93416.82016.90117.33717.41518.12918.117
HealthHlth12.25711.24012.19513.31514.14514.55915.944
HealthMedEq15.09113.96315.09116.87316.79518.94617.974
ManufacturingAero16.33314.90315.91418.33919.15820.64722.643
ManufacturingAutos17.12115.66916.71418.52420.19721.46422.892
ManufacturingBoxes13.69712.24413.52115.47615.64020.00522.820
ManufacturingElcEq17.72916.07717.32919.73320.46923.81025.480
ManufacturingFabPr11.94910.65711.53513.54715.45316.11418.031
ManufacturingGuns9.2368.1558.8589.97211.30913.89214.047
ManufacturingLabEq19.11517.75518.60720.77421.53023.38227.771
ManufacturingMach25.35323.13425.55427.71328.41231.28730.476
ManufacturingPaper19.57718.31419.62220.70421.70625.18026.474
ManufacturingRubbr15.76314.01515.28718.20719.32722.15426.683
ManufacturingShips10.1509.0429.78211.33212.20316.39617.330
ManufacturingSteel18.72317.53818.43920.51320.90322.97123.042
MoneyBanks23.90223.09723.24824.51524.89927.02532.536
MoneyFin28.06524.60627.84132.58134.21535.43042.598
MoneyInsur22.79121.57922.82024.65225.69025.43727.779
MoneyRlEst13.79212.19213.42815.35116.48218.42423.825
OtherBldMt23.16020.72923.10225.69227.50928.18730.079
OtherBusSv29.79226.77129.04434.17436.57943.39749.071
OtherCnstr16.44614.35216.45118.83520.24622.90522.818
OtherFun12.92011.57913.05814.07115.28216.89220.068
OtherGold2.1721.7521.8592.9483.1472.9952.716
OtherMeals15.40614.14514.90216.21618.57420.73021.741
OtherOther19.08517.70218.32820.31621.83424.97928.458
OtherTrans20.57219.16719.70522.06723.51926.41423.600
ShopsPerSv14.24112.78113.98915.64217.17819.80722.745
ShopsRtail21.00220.07120.62921.61524.24624.58525.547
ShopsWhlsl24.82622.53824.08627.63230.98136.59437.185
TelecommunicationsTelcm17.78417.63416.97918.25418.72320.46421.987
UtilitiesUtil17.38517.87318.22218.48617.15617.95220.056

### Table 16.

Average t-statistics for the Mkt risk parameter.

SectorIndustryStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
ChemicalsChems0.06640.04080.09030.11040.08820.02860.0412
Consumer DurablesHshld−0.2695−0.2885−0.2902−0.2755−0.1750−0.1320−0.2072
Consumer Non-DurablesAgric0.37580.40000.33460.43650.33170.43730.4856
Consumer Non-DurablesBeer−0.1894−0.1742−0.1779−0.2261−0.2481−0.1270−0.1546
Consumer Non-DurablesBooks0.30680.31460.30860.26640.25200.20680.1749
Consumer Non-DurablesClths0.45960.46000.46680.44700.39150.42890.4416
Consumer Non-DurablesFood−0.0900−0.0347−0.0801−0.1009−0.1251−0.1603−0.2084
Consumer Non-DurablesSmoke−0.2571−0.2403−0.2177−0.3084−0.2380−0.2543−0.2380
Consumer Non-DurablesSoda−0.1360−0.0748−0.1717−0.2282−0.22550.0180−0.0552
Consumer Non-DurablesToys0.29470.23370.27280.30090.39840.60790.3526
Consumer Non-DurablesTxtls0.61640.61250.58590.62080.68220.53050.5190
EnergyCoal0.51920.52250.49460.57780.75560.51830.2613
EnergyMines0.42350.40230.42910.47100.39210.40070.7195
EnergyOil−0.2021−0.2319−0.1798−0.1753−0.1043−0.0953−0.0733
HealthDrugs−0.2174−0.1567−0.2341−0.2440−0.2329−0.2875−0.3023
HealthHlth0.59920.61070.61460.61260.49970.54900.6607
HealthMedEq0.09190.10460.11110.04940.07390.09740.1784
ManufacturingAero0.10870.07590.11820.16030.17750.11780.1247
ManufacturingAutos−0.0033−0.1007−0.00180.09750.14870.11890.2935
ManufacturingBoxes0.11440.16140.11090.12480.11370.0077−0.1318
ManufacturingElcEq0.08350.07990.01820.10730.16360.14780.1187
ManufacturingFabPr0.71370.73140.70140.75440.72370.69210.6956
ManufacturingGuns0.0159−0.0055−0.03230.09500.03780.07230.0229
ManufacturingLabEq0.30920.26810.28570.29680.32110.46890.5188
ManufacturingMach0.35270.34190.31530.37990.41940.41270.3551
ManufacturingPaper0.10910.13840.11460.10110.01610.0108−0.0158
ManufacturingRubbr0.53630.53320.49600.56630.52630.53070.3955
ManufacturingShips0.34840.35040.31060.42580.28080.29840.3986
ManufacturingSteel0.43300.40860.46140.45480.44830.43270.7188
MoneyBanks0.06510.07400.04250.02190.0581−0.05530.0099
MoneyFin0.23490.24590.23140.17810.17350.17650.1781
MoneyInsur0.11040.14170.15840.0839−0.0180−0.0752−0.0715
MoneyRlEst0.73550.75250.73100.71340.65230.71500.8606
OtherBldMt0.36070.35960.35740.34100.32690.39680.2694
OtherBusSv0.48790.51110.49510.45730.45390.40400.4887
OtherCnstr0.60690.58310.67860.59670.63440.56710.5042
OtherFun0.26880.22970.28010.28860.20840.42700.5197
OtherGold0.43730.33360.38020.64020.58360.25350.7318
OtherMeals0.15800.11990.16920.17840.11750.22470.2498
OtherOther0.30380.32890.35750.27650.31560.17690.2268
OtherTrans0.26980.26990.27860.27470.29050.38220.2701
ShopsPerSv0.58840.63410.61420.47370.41570.45310.7126
ShopsRtail0.05310.05860.07210.05060.00310.03460.0209
ShopsWhlsl0.41610.41500.38760.42910.38490.47200.4194
TelecommunicationsTelcm−0.1961−0.1676−0.2229−0.2066−0.2111−0.2495−0.0907
UtilitiesUtil−0.03150.0290−0.0185−0.0309−0.1474−0.2308−0.3373

### Table 17.

Average parameters for SMB by industry.

SectorIndustryStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
ChemicalsChems0.80490.49261.02851.19410.94900.32950.7415
Consumer DurablesHshld−3.5220−3.3883−3.5238−3.5921−2.8983−2.1829−4.8518
Consumer Non-DurablesAgric2.51052.29232.17192.84552.63893.41215.8980
Consumer Non-DurablesBeer−1.7689−1.6097−1.7492−2.0344−2.1755−0.8634−2.7206
Consumer Non-DurablesBooks3.26993.06583.06003.02472.85982.33272.7036
Consumer Non-DurablesClths4.64594.25924.40684.49064.37274.95566.6808
Consumer Non-DurablesFood−0.7780−0.2173−0.7536−0.6514−1.4482−1.7753−2.1650
Consumer Non-DurablesSmoke−1.9182−1.6316−1.6903−2.2337−1.8367−1.8603−3.0293
Consumer Non-DurablesSoda−1.2139−0.7835−1.2947−1.6556−2.0505−0.5054−1.6006
Consumer Non-DurablesToys2.27611.80552.10912.32713.25395.59062.8274
Consumer Non-DurablesTxtls5.86025.32745.36665.88746.91266.03507.7287
EnergyCoal2.16502.07551.95262.38963.60662.74362.3730
EnergyMines3.11072.77703.08123.28622.96622.75825.6042
EnergyOil−2.0905−2.3782−1.7681−1.6474−1.0455−0.8614−0.0473
HealthDrugs−2.4303−1.6248−2.5461−2.7498−2.7144−3.3888−4.1032
HealthHlth4.04723.86943.90823.83693.83144.35676.1027
HealthMedEq1.04141.18601.15650.57780.66321.12812.7113
ManufacturingAero0.74590.29130.66061.18661.62131.33202.0286
ManufacturingAutos0.0875−0.50220.16850.84370.86940.80373.2068
ManufacturingBoxes0.99611.27500.94641.03940.8415−0.1866−2.0069
ManufacturingElcEq1.30741.19080.79941.61852.10092.33641.8252
ManufacturingFabPr5.15384.72424.93145.47616.33646.15547.7016
ManufacturingGuns−0.1197−0.4346−0.43610.53180.28400.51030.2687
ManufacturingLabEq3.48623.02383.17423.57873.78785.35678.5789
ManufacturingMach4.92374.62584.35224.91865.87875.62685.8431
ManufacturingPaper1.32131.55651.28041.02020.54310.04060.6256
ManufacturingRubbr5.44364.83954.82415.98506.15957.24277.7446
ManufacturingShips2.21702.00872.03312.68192.07962.04213.0639
ManufacturingSteel3.85063.58804.02023.93974.22763.98856.3386
MoneyBanks0.85551.10920.51800.22030.4598−1.4619−0.9630
MoneyFin4.08073.86413.92873.54563.39652.99592.9406
MoneyInsur1.58321.96822.03591.1199−0.1656−0.8606−1.7632
MoneyRlEst5.99445.49565.73255.91636.10717.197110.0976
OtherBldMt5.30715.07765.01104.76774.95015.80934.5240
OtherBusSv8.65178.17848.30388.41589.08639.559213.4606
OtherCnstr5.07534.48265.34464.89555.87436.68566.7519
OtherFun1.90421.61881.98482.14921.57323.73084.8763
OtherGold1.40680.95381.15722.01912.34641.10642.9292
OtherMeals1.55601.34601.60731.52250.88092.61342.4165
OtherOther3.61193.27943.67553.21384.49553.26504.9368
OtherTrans3.06502.90893.00213.02663.41034.81663.8428
ShopsPerSv5.08625.12744.95624.24333.83344.62789.3979
ShopsRtail0.80200.85290.94810.72170.03970.3771−0.9400
ShopsWhlsl6.37875.73855.64786.63117.235810.246610.6375
TelecommunicationsTelcm−2.4523−1.9666−2.4536−2.4007−3.0064−3.6550−1.8704
UtilitiesUtil−0.22031.12650.0408−0.2535−1.9483−3.0557−5.3771

### Table 18.

Average t-statistics for the size parameter, SMB.

SectorIndustryStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
ChemicalsChems0.19270.14490.18890.22780.26950.24170.3472
Consumer DurablesHshld−0.2313−0.2352−0.2307−0.2223−0.2604−0.3878−0.2267
Consumer Non-DurablesAgric0.0095−0.0165−0.01580.09920.0344−0.09040.0845
Consumer Non-DurablesBeer−0.2143−0.2016−0.1889−0.2460−0.2633−0.2624−0.0864
Consumer Non-DurablesBooks0.08620.09810.09330.09870.05680.07690.2010
Consumer Non-DurablesClths0.11220.14210.11340.09990.1115−0.0405−0.0016
Consumer Non-DurablesFood−0.0502−0.0040−0.0396−0.1035−0.1418−0.0782−0.0692
Consumer Non-DurablesSmoke−0.1436−0.1884−0.1401−0.1125−0.0478−0.0634−0.0241
Consumer Non-DurablesSoda−0.1529−0.1422−0.1075−0.2756−0.2496−0.2288−0.0798
Consumer Non-DurablesToys−0.1494−0.1311−0.1945−0.2309−0.0376−0.0763−0.1821
Consumer Non-DurablesTxtls0.31580.32600.33530.34100.15870.13910.4091
EnergyCoal0.50110.35910.55680.51430.60190.80630.8292
EnergyMines0.38040.31160.42160.42450.52550.39700.5823
EnergyOil0.45140.39390.48990.54670.53380.52980.5488
HealthDrugs−0.5554−0.5052−0.5491−0.5895−0.5725−0.6042−0.7282
HealthHlth−0.1292−0.0484−0.1273−0.1609−0.2080−0.1178−0.2578
HealthMedEq−0.3412−0.2495−0.3072−0.4346−0.4019−0.5114−0.4813
ManufacturingAero0.15640.14990.16670.19440.22850.03750.1123
ManufacturingAutos0.51130.51980.49650.52620.50370.48320.5802
ManufacturingBoxes0.09930.14720.07300.06430.11480.0842−0.1027
ManufacturingElcEq−0.0827−0.0828−0.1275−0.0159−0.1266−0.2090−0.1646
ManufacturingFabPr0.28000.35230.22800.20610.24660.18130.1398
ManufacturingGuns0.18310.23540.11390.17120.11300.09790.2811
ManufacturingLabEq−0.3127−0.2836−0.2694−0.3621−0.3496−0.4837−0.5052
ManufacturingMach0.13580.11260.12920.16610.14700.12980.2192
ManufacturingPaper0.22910.22750.22720.25150.26420.12780.3562
ManufacturingRubbr0.20290.24760.21630.18760.15100.04270.0903
ManufacturingShips0.20850.20720.20720.25230.19290.16420.0738
ManufacturingSteel0.64770.59660.65580.71830.78110.58440.7460
MoneyBanks0.57080.53300.55730.56810.56180.60510.6895
MoneyFin0.35450.34180.33030.32660.35300.31870.3266
MoneyInsur0.32190.32460.32570.29040.31730.34640.3412
MoneyRlEst0.32270.32100.30640.25610.23630.30580.4090
OtherBldMt0.21650.16790.24570.25840.23890.14830.2445
OtherBusSv−0.02870.0044−0.0173−0.0309−0.0591−0.0452−0.1486
OtherCnstr0.33170.36050.37200.28810.26870.44980.5322
OtherFun−0.1184−0.0801−0.1530−0.1116−0.1949−0.2539−0.0978
OtherGold0.13580.01250.08520.21770.28510.27290.7532
OtherMeals−0.2193−0.2227−0.2296−0.2285−0.2175−0.3029−0.2790
OtherOther0.03860.03650.05580.10120.00310.09100.0652
OtherTrans0.26080.27830.25570.26300.26730.20690.1664
ShopsPerSv0.01060.07540.0059−0.06640.0108−0.0061−0.0195
ShopsRtail−0.1028−0.0794−0.0716−0.1005−0.1233−0.2047−0.3034
ShopsWhlsl0.05100.07980.05770.05030.0075−0.0043−0.0177
TelecommunicationsTelcm0.25250.30530.27490.22560.19550.27420.0969
UtilitiesUtil0.41790.37180.42170.43490.46680.51260.3949

### Table 19.

Average parameters for HML by industry.

SectorIndustryStandardScale 1Scale 2Scale 3Scale 4Scale 5Scale 6
ChemicalsChems1.58071.03051.50222.09402.74002.96364.0330
Consumer DurablesHshld−2.3809−2.0684−2.3437−2.4775−3.7354−5.6384−4.9896
Consumer Non-DurablesAgric0.0740−0.11570.00240.48920.3488−0.5291−0.0022
Consumer Non-DurablesBeer−1.5311−1.3189−1.5000−1.8565−2.1101−2.3938−1.5355
Consumer Non-DurablesBooks0.91440.92470.96701.16100.57731.13212.4769
Consumer Non-DurablesClths0.94670.97350.97280.99991.21570.37390.5315
Consumer Non-DurablesFood−0.41120.0323−0.3587−0.8661−1.6998−0.7677−0.8502
Consumer Non-DurablesSmoke−0.9420−1.0478−0.9154−0.9083−0.7504−0.5228−0.6924
Consumer Non-DurablesSoda−0.8866−0.7657−0.7021−1.2258−1.5206−1.9345−2.9555
Consumer Non-DurablesToys−0.6113−0.3523−0.7208−1.2232−0.2576−0.4538−1.7159
Consumer Non-DurablesTxtls2.33412.07922.35352.74701.82222.07184.8854
EnergyCoal1.43930.94621.55781.73792.41232.63593.6478
EnergyMines2.04261.54052.24042.54773.31982.58473.8822
EnergyOil2.65812.31332.79923.20253.91424.33904.3149
HealthDrugs−5.0907−4.3717−4.8950−5.6002−6.1844−6.6855−8.9212
HealthHlth−0.6160−0.1308−0.6923−0.9001−1.3981−0.7229−2.5513
HealthMedEq−2.7898−1.9996−2.4451−3.7228−4.1971−5.7059−5.6987
ManufacturingAero0.96480.67731.09581.54361.99280.62221.5235
ManufacturingAutos3.28352.90803.18793.95044.12884.53405.8731
ManufacturingBoxes0.60100.77890.58980.47880.72800.3423−2.5781
ManufacturingElcEq−0.3872−0.3341−0.41230.0537−1.0260−2.1100−1.3577
ManufacturingFabPr1.47241.63841.15171.21851.62451.36011.6359
ManufacturingGuns0.66230.71060.37660.91480.61220.69421.9906
ManufacturingLabEq−2.2409−1.9613−1.9003−2.3894−3.1570−4.9967−6.9546
ManufacturingMach1.42611.10521.55131.79531.61901.29272.5047
ManufacturingPaper1.89171.61921.92792.35442.42261.28354.0063
ManufacturingRubbr1.56291.67631.58351.56721.48130.58141.0477
ManufacturingShips1.11881.00801.11441.43061.55711.24720.2335
ManufacturingSteel4.43223.78564.36005.32716.15785.27266.4820
MoneyBanks6.53296.15306.14356.65986.74977.940510.1990
MoneyFin4.47943.90674.22824.81005.57395.19446.3825
MoneyInsur3.86573.74643.83723.86794.08884.64345.1487
MoneyRlEst2.24492.03992.19461.89822.08442.77025.0913
OtherBldMt2.32641.76882.48672.89822.88962.10003.7157
OtherBusSv−0.5466−0.1084−0.4198−0.5681−1.3059−1.3618−4.3800
OtherCnstr2.11022.08122.22551.92961.95794.02805.2924
OtherFun−0.5212−0.2189−0.7152−0.4888−1.3764−1.9931−1.2353
OtherGold0.1164−0.1277−0.07200.37510.75300.32731.9220
OtherMeals−1.5515−1.3257−1.5194−1.7373−2.2743−3.2196−3.3617
OtherOther0.46630.51290.58660.73050.08571.08492.0143
OtherTrans2.19812.04022.15322.56882.79352.55441.9368
ShopsPerSv0.28210.58630.2316−0.12870.21140.1905−0.2324
ShopsRtail−1.2744−1.0379−0.9744−1.3434−1.8573−2.4995−3.1044
ShopsWhlsl0.37270.55340.47430.4695−0.2664−0.2654−1.1328
TelecommunicationsTelcm2.06162.08062.12541.92102.26033.39762.7911
UtilitiesUtil4.82744.38224.90515.30885.59516.56095.8998

### Table 20.

Average t-statistics for the book-to-market parameter, HML.

## References

1. 1. Banz R. The relationship between return and market value of common stocks. Journal of Financial Economics. 1981;9(1):3-18
2. 2. Berger T, Fieberg C. On portfolio optimization. The Journal of Risk Finance. 2016;17(3):295-309
3. 3. Bollerslev T, Engle RF, Wooldridge JM. A capital asset pricing model with time-varying covariances. Journal of Political Economy. 1988;96(1):116-131
4. 4. Daubechies I. Ten Lectures on Wavelets. CBMS-NSF Lecture Notes nr. 61. SIAM; 1992
5. 5. Fama E, French KR. The cross-section of expected stock returns. Journal of Finance. 1992;47(2):427-465
6. 6. Fernandez V. The international CAPM and a wavelet-based decomposition of value at risk. Studies in Nonlinear Dynamics & Econometrics Article 4 ser. 9.1. 2005
7. 7. Gencay R, Faruk S, Brandon W. Systematic risk and timescales. Quantitative Finance. 2003;3:108-116
8. 8. Gencay R, Faruk S, Brandon W. Multiscale systematic risk. Journal of International Money and Finance. 2005;24:55-70
9. 9. Gencay R, Faruk S, Brandon W. An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. New York: Academic Press; 2010
10. 10. In F, Kim S. The relationship between Fama-French three risk factors, industry portfolio returns, and industrial production, Francis. 2006. Available at SSRN: http://ssrn.com/abstract 891567
11. 11. In F, Kim S. A note on the relationship between Fama-French risk factors and innovations of ICAPM state variables. Finance Research Letters, Francis. 2007;4(3):165-171
12. 12. In F, Kim S, Faff R. Explaining mispricing with Fama French factors: New evidence from the multiscaling approach. Applied Financial Economics, Francis. 2010;20(4):323-330
13. 13. In F, Kim S. Portfolio allocation and the investment horizon: A multiscaling approach. Quantitative Finance, Francis. 2010;10(4):443-453
14. 14. In F, Kim S. Investment horizon effect on asset allocation between value and growth strategies. Economic Modelling, Francis. 2011;28:1489-1497
15. 15. Lintner J. Security prices, risk, and maximal gains from diversification. The Journal of Finance. 1965;20(4):587-615
16. 16. McNevin B, Nix J. The beta heuristic from a time/frequency perspective: A wavelet analysis of the market risk of sectors. Economic Modeling. 2017. Forthcoming
17. 17. Mossin J. Equilibrium in a capital asset market. Econometrica. 1966;34(4):768-783
18. 18. Percival DB, Walden AT. Wavelet Methods for Time Series Analysis. New York: Cambridge University Press; 2000
19. 19. Rosenberg B, Reid K, Lanstein R. Persuasive evidence of market inefficiency. The Journal of Portfolio Management. 1985;11(Spring, 3):9-16
20. 20. Rua A, Nunes LC. A wavelet based assessment of market risk: The emerging markets case. The Quarterly Review of Economics and Finance. 2012;52:84-92
21. 21. Sharpe W. Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance. 1964;19(3):425-442
22. 22. Trimech A, Kortas H, Benammou A, Benammou S. Multiscale Fama? French model: Application to the French market. The Journal of Risk Finance. 2009;10(2):179-192

## Notes

• Brazil, Chile, Mexico, Indonesia, South Korea, Malaysia, and Thailand.
• ICAPM for two countries Eri−r=β1covrirw+β2covris, where ri = returns for domestic asset, rw = returns for world portfolio, s is the percent change in the exchange rate for domestic and foreign currency.
• Their application is to Emerging Markets.
• Book-to-market is defined and total assets less total liabilities.
• http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
• See Chapter 4 of Gencay et al. [9] for additional detail.
• Percival and Walden provide a comparison of DWT and MODWT smooths for various filters which shows that MODWT MRAs are less sensitive to the filter type than DWT MRAs. See pp. 195–200 in Percival and Walden for a discussion on the practical considerations of the MODWT.
• Industry level parameter estimates and t-statistics for the alternative filters are available from the authors upon request.

Written By

Bruce D. McNevin and Joan Nix

Submitted: October 8th, 2017 Reviewed: January 18th, 2018 Published: October 3rd, 2018