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 <sup>~</sup>h1, and a low pass scaling filter <sup>g</sup>~<sup>1</sup> are applied to obtain wavelet coefficients w~<sup>1</sup> , and scaling coefficients v~<sup>1</sup> . In the second step of the pyramid, the original data series xt is replaced by <sup>v</sup>~<sup>1</sup> which is passed a high pass filter <sup>~</sup>h<sup>2</sup> and a low pass filter <sup>g</sup>~<sup>2</sup> to obtain wavelet and scaling coefficients, w~<sup>2</sup> , and v~<sup>2</sup> , 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).

<sup>6</sup> See Chapter 4 of Gencay et al. [9] for additional detail.

## 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.<sup>7</sup>

#### 3.2. Model specification

3. Methodology

88 Wavelet Theory and Its Applications

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

6

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 <sup>~</sup>h1, and a low pass scaling filter <sup>g</sup>~<sup>1</sup> are applied to obtain wavelet coefficients w~<sup>1</sup> , and scaling coefficients v~<sup>1</sup> . In the second step of the pyramid, the original data series xt is replaced by <sup>v</sup>~<sup>1</sup> which is passed a high pass filter <sup>~</sup>h<sup>2</sup> and a low pass filter <sup>g</sup>~<sup>2</sup> to obtain wavelet and scaling coefficients, w~<sup>2</sup> , and v~<sup>2</sup> , 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,

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

. 6

while the Discrete Wavelet Transform (DWT) can only be applied to series of size 2J

evaluate cover 2–4 days (J = 1) to 64–128 days (J = 6).

See Chapter 4 of Gencay et al. [9] for additional detail.

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

$$\begin{split} r\_{\text{it}}\left(\lambda\_{\text{j}}\right) - r\_{\text{f}}\left(\lambda\_{\text{j}}\right) &= a\_{\text{i}}\left(\lambda\_{\text{j}}\right) + \beta\_{\text{i}}\left(\lambda\_{\text{j}}\right) \* \left(\text{RM}\_{\text{t}}\left(\lambda\_{\text{j}}\right) - \text{RF}\_{\text{t}}\left(\lambda\_{\text{j}}\right)\right) \\ &+ \beta\_{\text{2i}}\left(\lambda\_{\text{j}}\right) \* \text{SMB}\_{\text{t}}\left(\lambda\_{\text{j}}\right) + \beta\_{\text{3}}\text{i}\*\left(\lambda\_{\text{j}}\right) \* \text{HML}\_{\text{t}}\left(\lambda\_{\text{j}}\right) + e\_{\text{ii}}\left(\lambda\_{\text{j}}\right) \end{split} \tag{3}$$

where <sup>λ</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup>�<sup>1</sup> , for j = 1, …,6. rit λ<sup>j</sup> � rf <sup>t</sup> <sup>λ</sup><sup>j</sup> is the excess return for industry portfolio i and time t, and scale j. RMtλj, RMtλj, SMBt λ<sup>j</sup> , and HMLt λ<sup>j</sup> 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.

<sup>7</sup> 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.

Figure 3. Three wavelet filters—DB(4), C(6), and LA(8).

We then rank the expected returns for that estimation period and assign a decile. The longshort 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.

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

Sector Standard Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment 1.184 1.215 1.162 1.138 1.139 1.149 1.147 Chemicals 1.083 1.049 1.086 1.102 1.136 1.176 1.084 Consumer Durables 0.849 0.874 0.831 0.822 0.764 0.769 0.857 Consumer Non-Durables 0.889 0.887 0.891 0.890 0.897 0.879 0.907 Energy 1.108 1.092 1.140 1.141 1.102 1.189 1.018 Health 0.955 0.963 0.986 0.960 0.945 0.948 0.866 Manufacturing 1.061 1.051 1.046 1.083 1.092 1.110 1.070 Money 1.091 1.062 1.086 1.123 1.133 1.109 1.214 Other 1.015 0.996 1.022 1.051 1.043 1.033 0.982 Shops 1.014 1.020 1.015 1.012 1.031 1.038 0.987 Telecommunications 0.888 0.941 0.881 0.867 0.830 0.849 0.805 Utilities 0.709 0.707 0.713 0.729 0.739 0.708 0.728

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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

include Chips, Software, Mines, Steel, Gold, and Lab. equipment.

Coal, Lab. Equipment, and Construction.

Table 4. Average Beta parameter by sector—LA(8).

#### 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.


Table 4. Average Beta parameter by sector—LA(8).

We then rank the expected returns for that estimation period and assign a decile. The longshort 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.

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 perfor-

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

4. Empirical findings

90 Wavelet Theory and Its Applications

Figure 3. Three wavelet filters—DB(4), C(6), and LA(8).

4.1. Parameter estimates

4.1.1. LA(8) filter

mance of the strategies during periods of recession.

between the standard beta and wavelet betas for scales 1 and 2.

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.

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.


Table 5. Average size parameter by sector—LA(8).


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.<sup>8</sup> 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

Sector Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment 1.211 1.162 1.142 1.141 1.146 1.139 Chemicals 1.051 1.086 1.103 1.132 1.168 1.105 Consumer Durables 0.870 0.835 0.825 0.773 0.780 0.848 Consumer Non-Durables 0.887 0.891 0.889 0.896 0.882 0.907 Energy 1.095 1.133 1.141 1.109 1.162 1.043 Health 0.964 0.983 0.961 0.947 0.948 0.878 Manufacturing 1.050 1.049 1.079 1.091 1.108 1.082 Money 1.063 1.086 1.119 1.131 1.117 1.202 Other 0.996 1.021 1.048 1.043 1.028 1.000 Shops 1.018 1.016 1.012 1.028 1.035 1.002 Telecommunications 0.937 0.883 0.864 0.835 0.847 0.814 Utilities 0.708 0.713 0.727 0.735 0.701 0.716

Sector Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment 1.210 1.162 1.141 1.141 1.143 1.137 Chemicals 1.051 1.086 1.103 1.131 1.164 1.109 Consumer Durables 0.869 0.835 0.824 0.774 0.781 0.849 Consumer Non-Durables 0.887 0.892 0.889 0.896 0.882 0.905 Energy 1.095 1.133 1.139 1.108 1.161 1.053 Health 0.964 0.983 0.961 0.946 0.949 0.879 Manufacturing 1.050 1.050 1.079 1.091 1.107 1.085 Money 1.063 1.086 1.119 1.130 1.116 1.201 Other 0.996 1.021 1.048 1.043 1.026 1.000 Shops 1.018 1.016 1.012 1.027 1.034 1.002 Telecommunications 0.937 0.883 0.863 0.835 0.848 0.814 Utilities 0.708 0.713 0.727 0.735 0.701 0.714

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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

filters.

Table 7. Average Beta parameter by sector—DB(4).

Table 8. Average Beta parameter by sector—C(6).

<sup>8</sup> Industry level parameter estimates and t-statistics for the alternative filters are available from the authors upon request.


Table 7. Average Beta parameter by sector—DB(4).


Table 8. Average Beta parameter by sector—C(6).

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

Table 6. Average book-to-market parameter by sector—LA(8).

Table 5. Average size parameter by sector—LA(8).

92 Wavelet Theory and Its Applications

8

In this section, we provide sector averages of parameter estimates for the Fama-French model based on two alternative filters.<sup>8</sup> 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)

Sector Standard Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment 0.175 0.092 0.182 0.242 0.312 0.412 0.463 Chemicals 0.066 0.041 0.090 0.110 0.088 0.029 0.041 Consumer Durables �0.270 �0.289 �0.290 �0.275 �0.175 �0.132 �0.207 Consumer Non-Durables 0.153 0.166 0.147 0.134 0.135 0.188 0.146 Energy 0.247 0.231 0.248 0.291 0.348 0.275 0.303 Health 0.158 0.186 0.164 0.139 0.114 0.120 0.179 Manufacturing 0.260 0.249 0.242 0.297 0.281 0.276 0.291 Money 0.286 0.304 0.291 0.249 0.216 0.190 0.244 Other 0.362 0.342 0.375 0.382 0.366 0.354 0.408 Shops 0.353 0.369 0.358 0.318 0.268 0.320 0.384 Telecommunications �0.196 �0.168 �0.223 �0.207 �0.211 �0.249 �0.091 Utilities �0.031 0.029 �0.018 �0.031 �0.147 �0.231 �0.337

Sector Standard Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment �0.661 �0.640 �0.686 �0.712 �0.672 �0.645 �0.583 Chemicals 0.193 0.145 0.189 0.228 0.270 0.242 0.347 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 �0.069 0.028 Energy 0.444 0.355 0.489 0.495 0.554 0.578 0.653 Health �0.342 �0.268 �0.328 �0.395 �0.394 �0.411 �0.489 Manufacturing 0.188 0.202 0.176 0.197 0.189 0.103 0.152 Money 0.392 0.380 0.380 0.360 0.367 0.394 0.442 Other 0.077 0.070 0.077 0.095 0.074 0.071 0.155 Shops �0.014 0.025 �0.003 �0.039 �0.035 �0.072 �0.114 Telecommunications 0.253 0.305 0.275 0.226 0.196 0.274 0.097 Utilities 0.418 0.372 0.422 0.435 0.467 0.513 0.395

Industry level parameter estimates and t-statistics for the alternative filters are available from the authors upon request.

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.

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


Table 9. Average size parameter by sector—DB(4).


Table 10. Average size parameter by sector—C(6).

the chapter focuses on the results for the LA(8) filter—a filter that is widely used in finance research employing wavelet methodology.

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

Sector Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment �0.640 �0.685 �0.700 �0.682 �0.634 �0.611 Chemicals 0.147 0.190 0.227 0.260 0.252 0.348 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 �0.061 0.022 Energy 0.366 0.475 0.500 0.552 0.570 0.624 Health �0.271 �0.327 �0.395 �0.388 �0.419 �0.465 Manufacturing 0.201 0.181 0.195 0.184 0.117 0.160 Money 0.381 0.382 0.364 0.375 0.393 0.447 Other 0.070 0.078 0.093 0.075 0.076 0.146 Shops 0.020 �0.002 �0.036 �0.038 �0.070 �0.085 Telecommunications 0.302 0.273 0.221 0.206 0.252 0.105 Utilities 0.376 0.420 0.435 0.468 0.500 0.385

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Sector Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment �0.640 �0.684 �0.701 �0.682 �0.637 �0.609 Chemicals 0.147 0.190 0.227 0.261 0.256 0.349 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 �0.061 0.021 Energy 0.365 0.476 0.498 0.553 0.573 0.636 Health �0.271 �0.328 �0.392 �0.389 �0.419 �0.464 Manufacturing 0.201 0.180 0.195 0.183 0.116 0.160 Money 0.381 0.381 0.363 0.373 0.391 0.442 Other 0.070 0.078 0.093 0.075 0.076 0.148 Shops 0.021 �0.002 �0.037 �0.040 �0.072 �0.090 Telecommunications 0.302 0.273 0.220 0.207 0.258 0.107 Utilities 0.376 0.420 0.435 0.469 0.502 0.391

for each of the models, even the standard model, is close to zero.

Table 11. Average book-to-market parameter by sector—DB(4).

Table 12. Average book-to-market parameter by sector—DB(4).

#### 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.

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Table 11. Average book-to-market parameter by sector—DB(4).


Table 12. Average book-to-market parameter by sector—DB(4).

the chapter focuses on the results for the LA(8) filter—a filter that is widely used in finance

Sector Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment 0.094 0.176 0.242 0.301 0.404 0.446 Chemicals 0.043 0.083 0.108 0.089 0.028 0.043 Consumer Durables �0.290 �0.287 �0.274 �0.186 �0.142 �0.209 Consumer Non-Durables 0.165 0.148 0.136 0.136 0.178 0.135 Energy 0.233 0.245 0.292 0.332 0.279 0.305 Health 0.185 0.164 0.140 0.120 0.125 0.156 Manufacturing 0.249 0.244 0.291 0.284 0.274 0.296 Money 0.303 0.291 0.255 0.222 0.188 0.231 Other 0.343 0.371 0.381 0.366 0.357 0.397 Shops 0.369 0.355 0.319 0.280 0.319 0.375 Telecommunications �0.172 �0.215 �0.213 �0.213 �0.223 �0.105 Utilities 0.027 �0.015 �0.037 �0.139 �0.217 �0.319

Sector Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Business Equipment 0.094 0.176 0.241 0.302 0.406 0.447 Chemicals 0.043 0.083 0.108 0.089 0.026 0.042 Consumer Durables �0.290 �0.288 �0.274 �0.187 �0.143 �0.211 Consumer Non-Durables 0.165 0.148 0.134 0.135 0.175 0.131 Energy 0.234 0.245 0.294 0.334 0.286 0.309 Health 0.185 0.164 0.140 0.121 0.122 0.147 Manufacturing 0.249 0.244 0.292 0.284 0.275 0.292 Money 0.303 0.291 0.253 0.221 0.190 0.232 Other 0.343 0.370 0.380 0.365 0.360 0.395 Shops 0.369 0.355 0.318 0.278 0.318 0.371 Telecommunications �0.172 �0.215 �0.214 �0.212 �0.221 �0.101 Utilities 0.027 �0.015 �0.039 �0.138 �0.215 �0.315

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.

research employing wavelet methodology.

Table 10. Average size parameter by sector—C(6).

Table 9. Average size parameter by sector—DB(4).

94 Wavelet Theory and Its Applications

4.2. Long-short strategy

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.


Table 13. Average 20-day cumulative returns for long-short strategy—LA(8).

#### 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 4–6 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 4–6

Recession Base Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Nov 1973–Mar 1975 0.78 1.35 3.94 1.28 0.22 5.97 0.10 Jan–July 1980 2.0 1.11 1.00 0.84 0.96 1.39 1.39 July 1981–Nov 1982 1.53 0.07 0.55 0.09 0.24 0.10 0.54 July 1990–Mar 1991 1.17 0.81 1.11 0.84 0.46 2.88 0.84 Mar 2001–Nov 2001 1.99 0.30 12.34 0.05 0.08 0.03 5.35 Dec 2007–June 2009 0.00 0.01 0.00 �0.41 0.11 0.00 2.35

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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

and Table 14).

Figure 6. Out-of-sample returns—long-short strategy LA(8).

Table 14. Cumulative out-of-sample returns during recessions—LA(8).

5. Conclusion

Figure 4. Out-of-sample returns—long-short strategy LA(8).

Figure 5. Out-of-sample returns—long-short strategy LA(8).


Table 14. Cumulative out-of-sample returns during recessions—LA(8).

Figure 6. Out-of-sample returns—long-short strategy LA(8).

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 4–6 and Table 14).
