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,<sup>5</sup> 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.

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.

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.

<sup>5</sup> http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data\_library.html.


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

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

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,<sup>5</sup> 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

The period of analysis cover five recessions, which are listed in Table 2. Our analysis of the

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

skewness varies across industries, but the returns for all industries are leptokurtotic.

performance of the long/short portfolios across scale focuses on these five recessions.

with potential recessionary periods.

2. Data discussion

84 Wavelet Theory and Its Applications

SMB portfolio is relatively large.

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data\_library.html.

5

Section 5 follows with our concluding comments.


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

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recessions.

Figure 1. Mkt returns and wavelet power.

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


Table 2. Recessions and duration in data sample.


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.

Figure 1. Mkt returns and wavelet power.

Sector Name Industry Mean Std.Dev Skewness Kurtosis Money Insur Insurance 0.0313 1.1682 �0.484 11.24 Money RlEst Real estate 0.0022 1.5172 �0.355 8.30 Other BldMt Construction materials 0.0282 1.2248 �0.306 7.41 Other BusSv Business services 0.0242 1.1153 �0.196 10.40 Other Cnstr Construction 0.0246 1.5836 �0.175 6.79 Other Fun Entertainment 0.0429 1.6688 0.342 20.74 Other Gold Precious metals 0.0244 2.3694 �0.018 16.42 Other Meals Restaurants, hotels, motels 0.0301 1.2684 0.299 16.98 Other Other Almost nothing 0.0030 1.4295 0.226 15.55 Other Trans Transportation 0.0273 1.2429 �0.161 11.68 Shops PerSv Personal services 0.0089 1.3192 �0.092 13.46 Shops Rtail Retail 0.0305 1.1679 �0.438 6.14 Shops Whlsl Wholesale 0.0245 1.0612 �0.528 9.46 Telecommunications Telcm Communication 0.0266 1.1191 �0.163 13.71 Utilities Util Utilities 0.0239 0.8743 0.012 21.07

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

Nov 1973–Mar 1975 16 Jan–July 1980 6 July 1981–Nov 1982 16 July 1990–Mar 1991 8 Mar 2001–Nov 2001 8 Dec 2007–June 2009 18

Table 2. Recessions and duration in data sample.

86 Wavelet Theory and Its Applications

Period Duration (mos)

MKt SMB HML

Mean 0.0253 0.0032 0.0171 Std. Dev. 1.0248 0.5435 0.5217 Skewness �0.5049 �1.0605 0.3507 Kurtosis 14.8116 23.2372 9.9377

Table 3. Summary statistics for model factors, daily data, July 1, 1967–September 29, 2017.

3.1. Selecting a filter

MRAs.<sup>7</sup>

3.2. Model specification

where <sup>λ</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup>�<sup>1</sup>

scale, j.

7

rit λ<sup>j</sup>

practical considerations of the MODWT.

� rf <sup>t</sup> <sup>λ</sup><sup>j</sup>

time t, and scale j. RMtλj, RMtλj, SMBt λ<sup>j</sup>

, for j = 1, …,6. rit λ<sup>j</sup>

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

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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

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.

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

∗ RMt λ<sup>j</sup>

<sup>þ</sup> <sup>β</sup>3i<sup>∗</sup> <sup>λ</sup><sup>j</sup>

, and HMLt λ<sup>j</sup>

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.

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

 � RFt <sup>λ</sup><sup>j</sup> 

∗HMLt λ<sup>j</sup>

<sup>þ</sup> eit <sup>λ</sup><sup>j</sup>

are the Fama-French factor for

is the excess return for industry portfolio i and

(3)

The greater the number of vanishing moments the smoother is the scale function.

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

<sup>þ</sup> <sup>β</sup><sup>i</sup> <sup>λ</sup><sup>j</sup>

� rf <sup>t</sup> <sup>λ</sup><sup>j</sup>

∗SMBt λ<sup>j</sup>

<sup>¼</sup> ai <sup>λ</sup><sup>j</sup>

þ β2<sup>i</sup> λ<sup>j</sup>

length L = 4, DB(4), and the Coiflet filter of length L = 6, C(6).

in research that applies wavelet methodology to finance.

Figure 2. SMB and HML, returns and wavelet power.
