Unique Factors. Yiyu Shen. Yexiao Xu. School of Management The University of Texas at Dallas. This version: March Abstract

Unique Factors By Yiyu Shen Yexiao Xu School of Management The University of Texas at Dallas This version: March 2006 Abstract In a multifactor model, individual stock returns are either determined by common risk factors that influence almost all stocks, or idiosyncratic risks that only affect severl stocks at most. With the deteriating explanatory power of the popular Fama and French three factor model, we focus on factors that may have impact only on certain groups of stocks in this paper. We call these factors Unique Factors. In particular, we present a simple approach both to extract unique factors from stock returns and to group stocks simultaneously. This allows us to have a parsimonious structure for individual stock returns with a very few number of groups. As a result, we find that a multifactor model with two common factors and two unique factors has superior explanatory power both in- and out-of-sample than models with only common factors including the Fama and French s (1992) factors and momentum factors. Moreover, in contrast to the declining explanatory power of common factors, the explanatory power of unique factors has increased over the past forty years. The address of the corresponding author is: Yexiao Xu, School Of Management, The University of Texas at Dallas, PO Box 688, Richardson, Texas 75080, USA; Email: yexiaoxu@utdallas.edu i

Unique Factors Abstract In a multifactor model, individual stock returns are either determined by common risk factors that influence almost all stocks, or idiosyncratic risks that only affect severl stocks at most. With the deteriating explanatory power of the popular Fama and French three factor model, we focus on factors that may have impact only on certain groups of stocks in this paper. We call these factors Unique Factors. In particular, we present a simple approach both to extract unique factors from stock returns and to group stocks simultaneously. This allows us to have a parsimonious structure for individual stock returns with a very few number of groups. As a result, we find that a multifactor model with two common factors and two unique factors has superior explanatory power both in- and out-of-sample than models with only common factors including the Fama and French s (1992) factors and momentum factors. Moreover, in contrast to the declining explanatory power of common factors, the explanatory power of unique factors has increased over the past forty years. Key Words: Explanatory Power, Heterogeneity, Idiosyncratic Volatility, Multifactor Model, Principal Component, Unique Factor. i

Introduction The classical CAPM theory suggests that individual stock returns are only determined by a single market factor. Recent empirical evidence from testing the CAPM model has shifted our attention to multifactor models. In practice, there are two approaches in applying a multifactor model the statistical factoring approach and the fundamental/macro factor approach. To a large extent, the statistical approach is motivated by the APT theory of Ross (1976). Starting from the assumption of a given linear factor structure of returns, the APT theory puts constraints between individual assets expected returns and the expected returns of the common factors. As shown by Chamberlain and Rothschild (1983), factors extracted using principal components (PC) analysis converge to a linear combination of the true underlying factors. In other words, the PC analysis can be used to extract the true factors ex post if we know the exact factor structure and have a large enough sample. In contrast, the fundamental/macro factor approach is a direct application of Merton s (1989) multifactor model, which implies that any factors that influence the growth of consumption should also price individual assets. Chen, Roll, and Ross (1986) have provided some evidence to support this approach. No matter which structure of a multifactor model is assumed, individual stock returns are either determined by common risk factors that influence almost all stocks, or idiosyncratic risks that only affect several stocks at most. Generally speaking, the statistical approach offers better in-sample fit but poor out-of-sample performance. In contrast, the fundamental factors can achieve better out-of-sample performance. This is the exact reason why recent literature has focused on fundamental factors, such as the Fama and French factors. 1 Therefore, there might be a middle ground between 1 Strictly speaking, Fama and French factors should be regarded as statistical factors since they are obtained by sorting individual stock returns according to firm characteristics. One can consider the 1

common factors and idiosyncratic risk factors. From an empirical perspective, Campbell, Lettau, Malkiel and Xu (2001) have shown that idiosyncratic volatilities have risen steadily while the market volatilities are stable in recent years. This phenomenon suggests that the explanatory power of a market model or the Fama-French model has deteriorated over time. At the same time, we observe that stock returns fluctuate more closely among their peers that engage in similar lines of business. These peers can, for example, belong to the same industries, have similar size, and/or have similar growth potential. In other words, there are increasing heterogeneity across groups and increasing homogeneity among firms within a group. It is thus reasonable to assume that there exist unique factors that only influence stocks in a particular group in addition to common factors that determine the returns of all stocks in the market. A multifactor model with both common factors and unique factors could describe the return structure more efficiently and precisely than models with only common factors. 2 There are evidences supporting the idea of unique factors. Both academic researchers and practitioners have long found factors related to certain groups of stocks according to characteristics. The two most distinguished examples are industry factors and country factors. But dividing stocks into different groups according to these characteristics has its limitations. On the one hand, there are hundreds of industries and countries, it is difficult to justify theoretically that each single industry or country has its own significant unique factors. When a statistical approach is applied to extract PC analysis as sorting stocks according to the characteristics of returns. Since firm characteristics are more stable than return characteristics, Fama and French factors usually have better out-of-sample properties. 2 Without considering the unique factors, we would have to consider an additional (J 1) K u number of factors if we treat all factors as common factors, where J and K u are number of groups and number of unique factors, respectively. Moreover, the extracted high order factors will represent a combination of unique factors from different groups, which makes it difficult to study the characteristics of unique factors. 2

factors for each industry or country, these factors are likely to be correlated with each other. On the other hand, grouping stocks according to these characteristics might not reflect the unique risk characteristics accurately. Many firms are multinational and/or multi-industrial firms. A pure SIC code classification might be misleading. It could also be true that several industries or countries share the same unique factors. More important, there are many other characteristics that can be potentially used to group stocks, such as the growth potential, firm size, social obligation, and so on. Therefore, a pure characteristics-based grouping method might be too complicated to capture the unique comovements in the stock markets. In order to effectively extract these distinctive risk factors, we need a more parsimonious approach to group stocks and to extract unique factors. In answering the challenge, we propose a statistical method to group stocks and to extract the corresponding unique factors simultaneously. Assuming there are unique factors pricing a particular group of stocks, our method is based on an iterative procedure: Starting from an arbitrary initial grouping, we extract unique factors for each group using the Maximum Explanatory Component (MEC) approach of Xu (2005). We then regroup each stock according to the differences in the explanatory power of each sets of unique factors for the stock. After obtaining the new groupings, we extract new sets of unique factors again. This procedure is continued until most individual stocks do not switch their group association. Similar to other statistical methods used to extract risk factors, our emphasis is not on the economic interpretation of these groups or these unique factors. However, we will try to establish some connection with, say Fama and French factors and industry association. We will also investigate the out-of-sample performance. Since our approach is parsimonious in terms of the number of groups, any significant gains in performance could not come from the restrictive structure. When applied to a sample of the US stocks over the sample period from 1965 to 2004, 3

we find encouraging results. For in-sample tests, our model of two common factors and two unique factors outperforms the four-factor models of Conner and Korajczyk (1988), Xu (2005), and Fama-French s (1993) three factors plus the momentum factor by 5% to 10% in terms of average adjusted R 2 s. For out-of-sample tests using the current group identifications and factor portfolio weights for the next period, our model still outperforms all other models used in this study. Moreover, in both the in-sample and the out-of-sample tests, stocks in each group are explained much better by their own unique factors than by the unique factors of other groups, which indicates that our method captures some of the heterogeneous group structure in stock markets. It is well documented that the degree of heterogeneity across all stocks has increased in the last 40 years for the U.S. stocks. This result does not necessarily suggest that stocks within the same group are less likely to be correlated with each other. Therefore it is interesting to investigate whether the explanatory power of unique factors has changed over time. We do a rolling regression analysis on U.S. stock markets for the sample period from 1950 to 2004 and find that this is indeed the case. Compared with the decreasing explanatory powers of the common factors, our unique factors are actually explaining more and more variation in individual stock returns within the same group. Another related issue is the increasing trend in the idiosyncratic volatility of stock returns during 1980s and 1990s. Since the measure of idiosyncratic volatilities is model dependent, the increasing trend documented by Campbell, Lettau, Malkiel and Xu (2001) might be less predominant when controlling for the unique factors. We do find supporting evidence. These results suggest heterogeneous groups and unique factors deserve more attention from both academia and practitioners. The rest of the paper is organized as follows. In Section 1, we motivate our approach through a numerical example, and lay out the details of our procedure in selecting groups and in estimating the unique factors. The in-sample and out-of-sample prop- 4

erties of the unique factors are then studied in Section 2 for NYSE/Amex/NASDAQ stocks. In Section 3, we provide further evidence of an increasing trend in both the explanatory power and the return volatilities explained by unique factors. Finally, we study how our unique factors are related to the Fama and French factors and 30- year bond returns, and the relation between our groupings of stocks and the industry groupings in section 4. Concluding comments are provided in Section 5. 5

1 Methodology and Data One can argue that the unique factors can be treated as common factors mathematically. It is true that we can capture the unique factors if allowing for enough number of factors in applying a statistical procedure to extract factors. Some of the factors will have zero loadings for some groups of stocks if there exist unique factors. The problem with this approach is that it is very difficult to tell whether the extracted high order factors represents unique factors or idiosyncratic returns of some stocks. In fact, these high order factors represent a combination of all unique factors, which makes interpretation difficult. Moreover, we could have used much fewer number of factors if we knew which stock is associated with which unique factors. For example, if there are three groups of stocks and each group of stocks have two unique factors, we have to extract four more factors when treating every factor as common factors. These points can be best illustrated through the following numerical example. Of course, we do not observe the grouping of stocks. A simple iterative procedure will be provided to deal with the issue later. 1.1 A Numerical Example Since our focus is on unique factors, not lose generality, we assume there are two unique factors r F 1 and r F 2, which determine returns of two groups of stocks respectively. For simplicity, there are two stocks in each group with the following return structure, Stock 1: r 1 = r F 1 + ɛ 1, ɛ 1 0 4 1 1 1 Stock 2: r 2 =2r F 1 + ɛ 2, ɛ 2 0 1 9 1 1 i.i.d.,. Stock 3: r 3 = r F 2 + ɛ 3, ɛ 3 0 1 1 4 1 Stock 4: r 4 =2r F 2 + ɛ 4, ɛ 4 0 1 1 1 9 6

The two unique factors r F 1 and r F 1 are independent with each other with an i.i.d. distribution of i.i.d.(0, 1). Clearly, we have also assumed the approximate factor structure of Chamberlain and Rothschild (1983), where the idiosyncratic risks can be correlated to some degree. This structure describes the actual stock returns better than the exact factor structure (with zero correlation among residuals). 3 Due to cross-sectional dispersion in idiosyncratic volatility, we apply the MEC approach of Xu (2005) to extract factors from individual stocks. In particular, when all four stocks are used, the first three MEC factors can be expressed in terms of original factors as, r mec 1a = 0.388r F 1 +0.388r F 2 + ɛ mec 1a, ɛ mec 1a N(0, 0.699), r mec 2a = 0.477r F 1 +0.477r F 2 + ɛ mec 2a, ɛ mec 2a N(0, 0.546), r mec 3a = 0.106r F 1 +0.106r F 2 + ɛ mec 3a, ɛ mec 3a N(0, 0.978), The first extracted factor is a linear combination of the two unique factors while the second extracted factor represents the difference of the two unique factors. This structure will not be altered if we also introduce a common factor. Therefore, it is difficult to study the behavior of unique factors alone. Ideally, the third extracted factor should only represent idiosyncratic returns. This is not the case here. When the grouping is known, we can extract the two unique factors separately. For example, using the first two stocks, the first unique factor can be extracted as, r mec 1b = 0.605r F 1 + ɛ mec 1b, ɛ mec 1b N(0, 0.634), r mec 2b = 0.096r F 1 + ɛ mec 2b, ɛ mec 2b N(0, 0.991). 3 If the exact factor structure is assumed, extracting two common factors from the four stock returns is equivalent to extracting the two unique factors from the first two stocks and the second two stocks, respectively. 7

The first MEC factor is indeed representing the first unique factor. Although the second MEC does not purely represent the idiosyncratic returns, the contamination is less severe. In fact, the variance of ɛ mec 2b is very close to one. In addition, we can study the goodness of fit for the two approaches. The following table compares factor loadings and the explanatory power (R 2 )usingthetwofactors extracted using all four stocks versus applying the unique factors extracted from their own groups of stocks. Using Common Factors Using Unique Factors ˆβ rmec1a ˆβrmec2a R 2 ˆβrmec1b ˆβrmec2b R 2 Stock 1 1.517-1.030 0.672 1.852 0.000 0.686 Stock 2 2.161-2.122 0.706 2.986 0.000 0.686 Stock 3 1.517 1.030 0.672 0.000 1.852 0.686 Stock 4 2.161 2.122 0.706 0.000 2.986 0.686 Average 0.698 0.696 StDev 0.322 1.668 0.567 0.567 Since the unique factor structure can be considered as the constrained common factor structure, unique factors should have lower explanatory power than those of common factors. In our example, however, the explanatory power using a single unique factor is almost as good as that using the two common factors. In this sense, the unique factor structure is parsimonious without sacrificing explanatory power. In addition, R 2 s are more evenly distributed across individual stocks when unique factors are used. Recent approaches to asset pricing tests have focused on cross-sectional regressions. The success of such tests usually relies on the cross-sectional dispersion of the regressors. As shown in the above table, the dispersion in beta for the first factor when using common factors is only 0.322. This is much smaller than either of the beta dispersions 8

of the unique factors. Therefore, the risk premium of the first factor will be estimated poorly relative to those of the unique factors. In addition, the loading of the second extracted factor can be negative or positive, which makes it difficult to interpret it as a pricing factor in the conventional sense (see Brown, Goetzmann, Grinblatt, 1997). 1.2 Assumptions About the Return Structure As motivated in the introduction, we modify the APT model to incorporate the unique factor structure by assuming there are J groups of stocks. Returns of the j-th group Stocks are linearly determined by K c number of common factors and K u number of unique factors. In particular, returns of the N j stocks in group j can be expressed as, R j = α j + β c F j c + β u F j u + ɛ j (1) E( F c ) = 0, E( F u )=0, (2) E( F c Fc ) = I K, E( F u Fu )=I K (3) E( ɛ j F c, F u ) = 0 (4) E( ɛ j ɛ j F c, F c ) = D j (5) where R j, ɛ j R N j T are matrices of the total returns and the idiosyncratic returns, respectively, across assets and over time, β c j R N j K c and β u j R N j K u are matrices of factor loadings with respect to the common pricing factors F c R Kc T and the unique factors F u R Ku T, respectively. α j R N j 1 is a vector of expected asset returns that satisfies the APT restrictions. 4 The variance-covariance matrix of residuals D needs not to be diagonal in order to allow for the approximate factor structure of Chamberlain and Rothschild (1983). In fact, we should only assume that the largest 4 If unique factors are treated as common factors given fixed number of groups, the current structure is a special structure of the classical APT model. Thus, the usual proof of the APT restrictions will carry through. 9

eigenvalue of D is bounded. When J = 1, the above model reduces to the classical APT model. Even when there are more than one group of stocks, we can still consider the above structure as a special case of the APT model, where stocks will have zero loadings on other groups unique factors. This suggests that factors extracted using principal components analysis (Chamberlain and Rothschild, 1983) or its variation (Xu, 2005) will also converge to a linear combination of the underlying factors. 1.3 Estimation Methods Similar to the issue of the number of underlying factors, the number of groups of stocks that have unique factors is unknown a priori. Therefore, our methodology starts with an assumption of J groups with K u unique factors in each group. In addition, there are K c common factors, which can be determined using various approaches including the Fama and French factors. 5 Conditioning on the common factors, the unique factors F u j are supposed to have maximum explanatory power for stocks within the j-th group if they are truly the unique factors. In other words, a stock should belong to group j if the stock returns are better explained by group j s unique factors than the unique factors of other groups. This idea motivates the following iterative procedure. Since idiosyncratic volatilities vary substantially across individual stocks, we will apply the MEC procedure of Xu (2005) in order to maximize the explanatory power of the extracted factors. Denote R R N T as the demeaned return matrix for all stocks, where N = J j=1 N j is the total number of stocks. Our iterative procedure will simultaneously divide stocks into J groups and extract K c common factors and K u unique factors for each group. This procedure follows the following steps: 5 In order to avoid the criticism that some of the explanatory power of unique factors may actually come from the common factors, we use the MEC approach of Xu (2005) to extract these common factors. This allows us to account for as much comovement among all stocks as possible. 10

1. Standardize each stock s returns by its standard deviation estimated from observations over time and denote it as R. 2. Extract the first K c largest eigenvectors from the cross-product matrix of C = 1 R R. Whenscaledby T 1, these eigenvectors will be the corresponding N 1 K c common factors ˆF c. 3. Run time-series regressions of each stock s returns on these common factors, and collect the residual returns. Denote the corresponding standardized residual returns as R. 4. Randomly divide the standardized residual returns into J groups, as R 1, R 2,, R J. 5. For the j-th group of stocks, extract the first K u largest eigenvectors from the matrix C j = 1 R N j R. Denote the corresponding unique factors as ˆF u 1 j. Repeat this process for each group. 6. Regress stock i s returns on the j-th set of unique factors ˆF u j, and record the coefficient of determination as R 2 i,j.stocki will be reassigned to the group j if R 2 i,j is largest, that is j = argmax j [R 2 i,j,j =1,,J]. 7. With the new grouping, go back to step 5 and repeat the unique factor extraction and reassignment of groups until no stock changes group membership. Applying this procedure, we are able to divide all stocks in the sample into J groups and extract K u unique factors for each group at the same time. For stocks in the same group, the group s own unique factors will have the largest explanatory power than unique factors of other groups, which reflects the definition of unique factors. Therefore, the linear model of equation (1) using these extracted common and unique factors is called Common and Unique Factor (CUF)model. 11

Many existing studies on this topic suggest that the majority variation in return can be captured by the first two common factors. As a result, we assume that there are two common factors in our empirical study. We also pre-specify the number of groups to be three with two unique factors in each group. Certainly, we do not know the true value of J, K c, and K u. One guideline in choosing these numbers is to keep these numbers as smaller as possible while allowing for a rich enough structure to describe the heterogeneity of individual equity returns. Previous studies such as Connor and Korajczyk (1988) and Jones (2001) have suggested that increasing the number of common factors beyond five has little additional explanation power. These results help us to keep the combined number of common factors and unique factors in our model at a comparable level. As a robust check we also test the specification with three groups, two common factor and three unique factors for each group. Results are comparable. 1.4 Data We use monthly return data on all U.S. stocks traded on NYSE/Amex stock exchanges between January 1965 and December 2004 from the CRSP data file. The 40 year sample is divided into eight five-year intervals since there are sufficient evidence suggesting time-varying volatility and covariance structure. Furthermore, the number of stocks available in our sample varies a lot over time. We will examine the in-sample performance of our CUF Factor model combining common factors and unique factors for each subsample period. For a robustness check, we also examine NASDAQ stocks over the sample period from 1975 to 2004. As a common practice, we discard stocks with prices below $1. In addition, we require stocks to have continuous return observa- 12

tions over the five-year subsample period. 6 Applying an iterative procedure outlined in Section 1.3, we are able to divide stocks into three groups and to extract both the common and the unique factors for each group efficiently. In order to be comparable to the existing literature on the issue of potential trends in R 2 s, we expand our sample period for NYSE/Amex stocks to start from 1950 in Section 3. The number of stocks in each group over time is presented in Panel A of Table 1 for NYSE/Amex stocks and in Panel B of Table 1 for NASDAQ stocks. These numbers have increased dramatically over our sample period a 60% increase for NYSE/Amex stocks and a 100% increase for NASDAQ stocks. The distribution of the number of stocks among the three groups is also close to uniform for both NYSE/Amex and NASDAQ stocks. Intuitively, the explanatory power of the unique factors of, say Group 1, will be high for a stock when the same stock has participated in extracting these unique factors. However, the marginal contribution of including an additional stock to the explanatory power of Group 1 s unique factors decreases with the number of stocks. Therefore, it might be better to include the stock in Group 2 or Group 3 instead, if the total number of stocks in these groups are relatively low. This mechanism will induce an even distribution across groups. However, mis-grouping should not be a concern if the unique factors of, say Group 1 has no explanatory power for stocks in other groups, and vice versa. In this sense, our approach of grouping stocks is superior to cluster analysis, which usually results in a very uneven distribution and is very sensitive to outliers. Insert Table 1 Approximately Here 6 For the same reason stated in section 2.2 of Jones (2001), we will ignore the case of missing observations. 13

2 Empirical Results For each stock in the sample, we fit returns to our four factor CUF model (including two common factors and two unique factors). For comparison, we also extract four factors using both the asymptotic components method of Connor and Korajczyk (1988) (CK) and the Maximum Explanatory Components (MEC) method of Xu (2005). In addition, we combine Fama and French s (1993) three factors (Size, Book-to-Market, and Market), and the momentum factor to form the so-called Fama-French four-factor model. 7 Due to the popularity of the Fama-French model, it can be used as a benchmark model. In this section, we investigate both the in-sample and the out-of-sample performance of our unique factors. 2.1 In-sample Performance We now investigate the in-sample explanatory power of the four competing models: our CUF factor model with two common factors and two unique factors, the CK fourfactor model, the MEC four-factor model, and the Fama-French four-factor model over the sample period from 1965 to 2004. Applying the procedure outlined in Section 1.3, we first divide all stocks into three groups and estimate the unique factors in each subsample period. For each non-overlapping five-year period, we then run a time-series regression using each models four factors and compute the average adjusted R 2 sof individual stocks in each group. In order to assess the explanatory power of unique factors, we also run similar time-series regressions for each stock using the three sets of unique factors individually and report the group wise average adjusted R 2 s. Table 2 shows the results for NYSE/Amex stocks, and Table 3 reports the results for NASDAQ stocks. 7 This model is sometimes labeled as the Carhart s (1997) four-factor model. The monthly returns of the four factors are obtained from Professor Kenneth French s website. 14

Insert Table 2 Approximately Here For NYSE/Amex stocks, column 7 of Table 2 indicates that the explanatory power of the Fama-French four-factor model has decreased substantially for all three groups of stocks. Comparing the adjusted R 2 s in the early 70s to those in the late 90s, there is almost a 50% drop. This phenomenon is consistent with Campbell, Lettau, Malkiel, and Xu s (2001) finding using a simple market model. The explanatory power has bounced back to a certain degree in the most recent subsample period for the third group of stocks. Similar trends exist when using either CK factors or MEC factors as shown in column 8 and column 10 of Table 2, respectively. In contrast, the explanatory power of our CUF model has only dropped by a third over the same sample period (see column 3 of Table 2). When compared across different models, our four-factor CUF model uniformly dominates the CK, themec, and the Fama- French four-factor models. For example the average adjusted R 2 s for the most recent subsample period are 33.7%, 29.0%, 23.2%, and 19.3% for our CUF,theMEC,the CK, and the Fama-French models, respectively. In fact, this is a universal pattern for the explanatory power of the four models over different subsample periods. Generally speaking, the four factor CUF model outperforms the Fama-French four-factor model by 5% to 15%. In addition, the MEC factors perform better than the CK factors, and the CK factors do a little better job than the Fama-French factors. It is equally important to study the explanatory power of unique factors and to see if they are indeed unique. A simple test would involve comparing the explanatory power of one group s unique factors for individual stock returns in the group to the explanatory power for other group s stocks. By definition, unique factors of a group should have high explanatory power for stocks in the group, and should have virtually no explanatory power for stocks from other groups. This is exactly the pattern shown 15

in column 4 through column 6 of Table 2. For example, the two unique factors of the first group have an average adjusted R 2 of 6.5% for stocks in the first group, and negative adjusted R 2 s for stocks in the other two groups. This is not an outcome by design since our procedure of grouping does not impose any orthogonality conditions for factors across different groups. Therefore, the unique factors do seem to capture the unique co-movement within each group only. The average adjusted R 2 s for the second and the third groups unique factors are 7.5% and 8.3%, respectively. As a comparison, if only the common factors are used as in the CK model, the additional explanatory for the high order factors (the third and the fourth factors) is limited (about 3%). For all subsample periods, these high order CK factors explain one or two groups of stock returns far better than other groups of stocks, which indirectly suggests a unique factor structure. It is also interesting to note that the explanatory power of unique factors has actually gone up in recent years, which is just the opposite of the common factors. We will further investigate this phenomenon in Section 3. Insert Table 3 Approximately Here For NASDAQ stocks shown in Table 3, the general patterns are very similar to those of NYSE/Amex stocks. In particular, the dominance of our CUF model continues to hold with an average R 2 of 8 percentage points lead to that of the Fama and French model. However, the explanatory power of any models is weaker now. For example, the average explanatory power of the Fama-French model itself is about 6 percentage points lower than that for NYSE/Amex stocks over the same sample period. What is more surprising is that the explanatory power in the most recent subsample period has bounced back to the level occured at the beginning of 80s. In fact, the explanatory power for NASDAQ stocks is now comparable to that for NYSE/Amex stocks. The explanatory power of the unique factors seems to be stable over different subsample 16

periods. Assuming two common factors may seem to be restrictive. If there is only one common factor, unique factors of one group should continue to have zero explanatory power for other groups of stocks. However, if there are indeed two common factors, but we only allow for one common factor in estimation, the unique factors of one group will also have explanatory power for other groups of stocks. This is indeed the case. When assuming one common factor and three unique factors for each of the three groups of stocks, we observe that the unique factors of one group have also helped to explain return variations in other groups. 8 This suggests that there are more than one common factor for NYSE/Amex stocks. Therefore, the structure of two common factors and two unique factors are maintained throughout the paper. Instead of extracting common factors, we can also use known risk factors as common factors. In particular, we use the market index and the liquidity factor of Pastor and Stambaugh (2003) as the common factors. Apparently the market index is used here as a proxy for the market risk factor. Liquidity is also included because of overwhelming evidence suggesting that it is indeed a risk factor. We do not use size or book-tomarket variables since they are not risk factor. 9 After computing residual returns from regressing individual stock returns on both the market and the liquidity factors, we apply similar approach to extract unique factors and construct the three groups. These results are reported in Table 4 for both NYSE/Amex and NASDAQ stocks. Insert Table 4 Approximately Here 8 Results are available from authors. 9 They are constructed according to the characteristics of a firm. An excellent discussion can be found in Cochrane (2001). 17

For NYSE/Amex stocks, the pattern in the explanatory power for both the Fama and French factors and the CUF factors are similar to that of the Table 2. 10 However, there still might be comovement after considering the two common factors as evident by the fact that the unique factors of one group have some explanatory power for stocks in other groups. This is especially the case in the most recent subsample period. We also observe that the difference between the explanatory power of the CUF model and the Fama-French model has widened from 8% to 10% when the fundamental risk factors are used. For NASDAQ stocks, the explanatory power of any model is generally lower than that for NYSE/Amex stocks. However, there are still 9% difference in the explanatory power between the CUF model and the Fama-French model. More important, the use of unique factors smoothes the explanatory power over time. These results further suggest that the existence of unique factors is robust to the choice of common factors. 2.2 Out-of-Sample Test It is well-known that factors extracted using statistical methods usually perform relatively well in-sample, but lack persistence performance out-of-sample. This is largely due to the fact that the variance-covariance structure varies over time. In contrast, the economic/fundamental factors possess good out-of-sample performance since firm characteristics are more stable over time. If our method of grouping stocks indeed captures the structure of heterogeneity across groups and of homogeneous movement within each group of individual stocks, we should expect to see the group structure being preserved out-of-sample. Therefore, it is useful to further investigate whether our model delivers good out-of-sample performance in a horse race. From a practical perspective, an application of unique factors in, say risk management, requires a construction of these factors ex ante. These constructed unique factors will have 10 Results in Table 2 and Table 4 are not exactly the same since the groupings are somewhat different. 18

large comovement with future returns of individual stocks only when they have good out-of-sample performance. We perform out-of-sample tests using a rolling method with a four-year factor portfolio formation period and a six-month out-of-sample testing period. We choose a six-month period in order to preserve the stability in the structure. From a practical perspective, rebalancing portfolios every six months does not seem to be often. In order to have sufficient observations to run a regression, we use six-month weekly return data for individual stocks. In particular, for each period we use the first 48 (four years) monthly returns to divided stocks into three groups and extract two common factors and two unique factors for each group using the iterative procedure outlined in Section 1.3. Applying the same group identifications and the factor portfolio weights to construct next six-month s weakly factor returns (both the common and the unique factors). We also construct the four corresponding out-of-sample CK factors of Connor and Korajczyk (1988) and the four MEC factors of Xu (2005). For the Fama-French four-factor model we simply use the data as obtained from Professor French s website. With these predicted factor returns, we compute the adjusted R 2 s from running time series regressions of each stock s weekly returns on the factors of different models. We then roll the whole testing period forward by six months to get the out-of-sample statistics for the next six months. For easy comparison, we average the adjusted R 2 s of each model for each five year subsample period. The results are reported in Table 5 for NYSE/Amex stocks and in Table 6 for NASDAQ stocks. Insert Table 5 Approximately Here For NYSE/Amex stocks, although the adjusted R 2 s are generally lower than those reported in Table 2 for in-sample tests, they are not entirely comparable since weekly returns are used here. In general, the in-sample performance of statistical methods 19

exceeds their out-of-sample performance, and R 2 s from high frequency returns are lower than those from low frequency returns. It is clear, however, the MEC factors of Xu(2005) have produced higher average R 2 than either the Fama-French four-factor model or the CK four-factor model for every period and in each group. This is because that the MEC approach explicitly discounts the influence of idiosyncratic volatilities which change a lot over time. To a large extend, our unique factor approach takes care of heterogeneity across groups. In some sense, CUF factor should outperform the MEC factors. This is indeed the case. When all the models are compared to our CUF model, our four factors even outperform the MEC factors by almost 1%. When compared to the benchmark (Fama-French) model, the differences are statistically significant at a 1% level (see the last column in Table 5). This is a strong indication that models with unique factors can explain return variations more efficiently. Note also the Fama-French factors have outperformed the CK factors in most of the subsample periods. Another important factor to consider is if the unique factors remain unique and whether the group structures are preserved out-of-sample. For comparison, we run three regressions using the two common factors and the first group s two unique factors (denoted as CUF 1 ), the two common factors and the second group s two unique factors (denoted as CUF 2 ), and the two common factors and the third group s two unique factors (denoted as CUF 3 ) for each stock. If the predicted factors are still unique outof-sample, we should expect to see, for example, that group one s own unique factors (CUF 1 ) should have larger explanatory power for stocks in group 1 than stocks in either group 2 or group3. This is exactly the case as shown in column six of Table 5. In fact, there are over 2% differences in R 2 s across groups. The same pattern occurs for unique factors from the second or the third groups (see columns 7 and 8 in Table 5). At the same time, if the group structures are preserved out-of-sample, for the same stock, 20

our four-factor model with its own unique factors should have the largest explanatory power compared with the four-factor model using other groups unique factors. For example, for group 1 s stocks, the adjusted R 2 s are higher under the column of CUF 1 than those under the columns of CUF 2 or CUF 3. The same things are true for the second and the third groups. Therefore, both the groupings and the unique factors for NYSE/Amex stocks seem to be preserved in short run. Insert Table 6 Approximately Here For NASDAQ stocks, the overall results are very similar to those for NYSE/Amex stocks. However, during the high-tech boom of the 90s, the explanatory power of any models is very low (between 11% and 14%). Since most high-tech firms are traded on NASDAQ, the evidence suggests that factor model does poorly during fast market condition change. Fortunately, our CUF model can gain over one percent in the explanatory power even under such condition. This gives another reason to focus on unique factors. Overall, the results in this section show not only our four-factor model (CUF) has better explanatory power out-of-sample, but also the groupings and the uniqueness of factors indeed have certain inherent structures. They are not statistical artifacts. 21

3 Historical Trend and Idiosyncratic Volatility Since the study by Campbell, Lettau, Malkiel and Xu (2001) s, many researchers have suggested that U.S. stock markets have become more and more heterogeneous in the past decade. We now have more classifications of industries. Many firms operate in many regions and countries. Even firms ages and financial conditions are becoming more diverse. Several decades ago only firms with a long and stable profit history were publicly traded. Nowadays, many companies with virtually no history of positive earnings go public. These observations suggest that if we consider the same set of common factors (or just the market factor as a special case) as source of risks, we expect the factors to explain less and less variation in stock returns over time. This conjecture is mirrored in the recently studies on idiosyncratic volatilities in the U.S. stock markets. In particular, Campbell, Lettau, Malkiel and Xu (2001) s have found that the idiosyncratic volatility of individual firms has been steady increasing over the past 40 years. In contrast, a piece of good or bad news about a firm spreads to other firms with similar characteristics quickly. Use of new technology makes it easier for a firm to introduce a similar product in respond to its competitors. The product development cycles become shorter and many firms offer similar products, which suggest intensified competition amongst a group firms. As a result, we see tighter fluctuations among firms with similar characteristics. While there is a decreasing trend in the explanatory power of common factors due to increases in heterogeneity across firms, we might also see that the explanatory power of unique factors increases over time as the preliminary evidence in Table 2 indicates. We take a closer look at the issue in this section. For a better estimation of the time-trend in the explanatory powers of various models and their corresponding idiosyncratic volatilities accurately, we adopt the rolling- 22

regression method proposed by Xu and Malkiel (2003). In particular, we use a 36-month rolling window in order to both preserve the dynamic properties of the estimated time series and the estimation efficiency. During each 36-month window, we first divide stocks into three groups and extract two common factors and two unique factors using the procedure described in Section 1.3. We use the grouping from last 36-month period as the starting point to estimate the groupings of the current period in order to ensure the stability of groupings. 11 Since measures of idiosyncratic volatility are model dependent and are essentially the risk that cannot be explained by factor models, we also extract both the four CK factors and the four MEC factors for the same sample period using all stocks. According to Xu and Malkiel (2003), each observation is weighted by a geometric declining weights in computing the volatility. For convenience, it is easy to see that the weighted residual variance can be obtained directly from regressions using transformed data. The adjusted R 2 s from such regressions will also be comparable. In particular, we first transform the time series of both individual stock returns and factor returns (including the intercept) with the same weights of ρ k/2 ( 1 ρ )(k =35,, 0) from the 1 ρ 18 (t 35)-th observations to the t th observation. In this study, we choose ρ =0.95. 12 We then run time series regressions of transformed individual stock returns on transformed factors (with transformed intercept) for various models to obtain the corresponding residual mean squared error (the idiosyncratic volatility measure), the total volatility, and R 2 directly. The time series pattern of R 2 s for different models can be seen from Figure 1. For 11 If there are new stocks added into the current sample, they are randomly allocated to the three groups with equal numbers at the beginning. 12 Using geometrically declining weights to compute the time-varying volatilities can approximate the GARCH(1, 1) volatility estimates. As shown in Xu and Malkiel (2003), the optimal weights depend on the persistence of the underlying volatility structure. Our conclusion does not change if we use ρ =0.9. 23

NYSE/Amex stocks as shown in the first panel of Figure 1, the explanatory power of the market model has declined dramatically from 40% in the early 50s to 20% in recent years. Such a declining trend in the explanatory power of either the Fama and French four-factor model or the CK four-factor model is less apparent. The R 2 has decreased from 50% in the 50s to 40% in recent years. In other words the performance gap between a multifactor model and the market model keeps widening, which suggests the increasing importance of a multifactor model in recent years. Of all the models we studied, the explanatory power of our four-factor CUF model is the highest although it also fluctuates around 55% till early 90s. Specifically, our CUF model outperforms both the Fama-French and the CK models by 5% to 10% in R 2 s. In addition, we note that the difference between our four-factor model and the Fama-French four-factor model shrinks from 1960 to 1990 and widens in both the 50s and recent years, which makes the declining trend less significant for our model. Insert Figure 1 Approximately Here The pattern in the explanatory power of the market model for NASDAQ stocks is very different as shown in the second panel of Figure 1. It rose from 20% in the late 70s to 35% in the early 90s, and then declined to 30% in recent years. The explanatory powers of the Fama-French four-factor model and the CK four-factor model are again generally on the same level. Both models offer a 10% increase in the explanatory power over the pure market model. At the same time, our four-factor model has outperformed the Fama-French model by 5% to 10% until early 90s. For the past several years, the gap widens again between our model and the other four-factor models. In order to see if the perceived pattern in the explanatory power reflects a time trend, or the persistence, or both in the underlying series, we perform statistical tests (see Panel A of Table 7). Since the conditional volatilities of stock returns are highly 24

persistent, we also find that the explanatory power (R 2 s) is very persistent. In fact, we have failed to reject the unit root hypothesis for all the R 2 series using the following testing model, y t = α + θ y t 1 + θ 1 y t 1 + θ 2 y t 2 + + θ 12 y t 12 + γ t, (6) where y t stands for various time series being tested. Of course, this failure in rejecting the unit-root hypothesis could simply due to structure breaks or regime switch. However, we can at least say that an increase or decrease in the explanatory power seems to have a long-term impact. It is also interesting to see that the γ estimates on the time trend are all negative for both NYSE/Amex and NASDAQ stocks, although most of them are insignificant. This is consistent with our observation from Figure 1. The downward trend for NYSE/Amex stocks using the market model is significant at a 95% level and is the largest among all models, which means that the market model is very inadequate in recent years. For robustness, we also use the PS trend test of Vogelsang (1998). This test is robust in the sense that it can accommodate both stationary and non-stationary residuals 13 Again all trend estimates are negative for NYSE/Amex stocks. But only the 10% confidence interval for the market model estimate close to reject the zero trend hypothesis. The results are much weaker for NASDAQ stocks. Insert Table 7 Approximately Here Given the fact that our four-factor model has experienced the smallest downward trend in the explanatory power, perhaps it is more important to study the explanatory power of the unique factors over time. For this purpose, we run the same rolling weighted regressions for each stock using the unique factors only. We plot the R 2 s 13 The trend estimates from this test is very different from that of equation (6) since the persistence is modeled through residuals. 25

from both using the unique factors and using the market factor for comparison in Figure 2 for NYSE/Amex stocks and in Figure 3 for NASDAQ stocks. In aggregate, the two unique factors can explain 15% more variation in individual stock returns on average for both NYSE/Amex and NASDAQ stocks in addition to the two common factors. This is a significant contribution especially during a time when the market model only explains 20% of variation in individual stock returns. Furthermore, there is an apparent upward trend in the explanatory power of the unique factors, especially for each of the individual groups of the NYSE/Amex stocks. However, the trend is less obvious for NASDAQ stocks. From the evidence, it is at least safe to say that there is no drop in the explanatory power of unique factors. Insert Figures 2 and 3 Approximately Here We can perform the same trend tests for the explanatory power of unique factors in Panel B of Table 7. Again we have failed to reject the unit-root hypothesis for both NYSE/Amex and NASDAQ stocks. The PS test indicates that there is a significant and positive trend in the explanatory power of unique factors for the second group of NYSE/Amex stocks. Although the trend estimates are positive for other groups of NYSE/Amex stock, they are statistically insignificant. For NASDAQ stock the results are mixed largely due to the fact that the explanatory power exhibits a U shape over time. The remaining question to ask is what drives the upward trend in the explanatory power of the unique factors. As a preliminary investigation, we will try to link the phenomenon to volatility instead of the fundament structure of the economy. An increase in the R 2 can be attributed to either a decrease in the total volatility of individual stocks or an increase in the volatility explained by the unique factors. Since the market volatility is stable and the idiosyncratic volatility has gone up, we can hypothesize 26