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asset reallocation_JFQA.doc

  1. 1. Do Aggressive Funds Reallocate Their Portfolios Aggressively? Kevin C.H. Chiang College of Business Administration Northern Arizona University Flagstaff, AZ 86011-5066 First draft: June 25, 2003 This draft: April 05, 2005 JEL Classification: D9; G11 Keywords: Asset allocation; dynamic choice; risk aversion; mutual funds * Correspondence: Kevin C.H. Chiang, College of Business Administration, Northern Arizona University, Flagstaff, AZ 86011-5066. Phone: (928) 523-4586, Fax: (928) 523-7331, E-mail: Kevin.Chiang@nau.edu. The author thanks Chuck Hamel, Ji-Chai Lin, Craig Wisen, and Thomas Zhou for helpful comments and suggestions. Do Aggressive Funds Reallocate Their Portfolios Aggressively?
  2. 2. Abstract This study examines pairs of asset allocation mutual funds that are controlled for all informational attributes, except for the level of risk aversion. Standard mean-variance models of portfolio choice suggest that the percentage rebalancing of common stocks in aggressive funds would be the same as that in conservative funds. However, the study finds the rebalancing of common stocks in aggressive funds to be disproportionally less intense. 2
  3. 3. Do Aggressive Funds Reallocate Their Portfolios Aggressively? I. Introduction More than ever before, investors delegate their portfolio decisions to professional fund managers. Today, mutual funds are a $7 trillion industry serving more than 90 million investors. As an important constituent of the capital market, the mutual fund industry has been extensively examined in a variety of aspects: performance, incentive structures, gaming behaviors, regulatory designs, to name just a few. However, the important issue of how mutual funds determine their asset allocation decisions is rarely investigated. This seems surprising because asset allocation is fundamental to the performance of an institutional portfolio (Brinson, Singer, and Beebower (1991)), and this motivates us to examine the role of risk preference on mutual funds’ asset allocations. Traditionally, asset allocation had been studied in the classic, one-period mean- variance framework of Markowitz (1952). Recently, the need for studying asset allocation in multi-period horizons has been growing. One catalyst for this need is the “asset allocation puzzle” of Canner, Mankiw, and Weil (1997). The authors document that popular advice on asset allocations to common stocks, bonds, and cash is inconsistent with the two-fund separation theorem in that the bonds to stocks ratio is lower (higher) for aggressive (conservative) investors. This finding is at odds with the well-known property of the static separation theorem that investment decisions are independent of financing decisions. There are three directions in the literature in which rationales are proposed to explain the asset allocation puzzle. First, asset allocation is examined in a positive 3
  4. 4. setting. Shefrin and Statman (2000) develop a behavioral portfolio theory in which multiple layers of investments are segmented from one another. They argue that popular advice is constructed as a pyramid: cash in the bottom (risk-free) layer, bonds in the middle (less risky) layer, and stocks in the top (more risky) layer. Because of this segmentation, the two-fund separation no longer holds and risk management is primarily achieved through mixing these layers. Therefore, the bonds to stocks ratio and the cash to stocks ratio can be increasing in risk aversion. Second, the usual assumption of no short-selling for establishing the separation theorem is relaxed. Elton and Gruber (2000) point out that if short sales are not allowed, the separation theorem would no longer hold. The efficient frontier is concave, bounded, and composed of linear combinations of adjacent efficient portfolios between corner portfolios. Since financing is impossible and the expected return on a portfolio is a linear combination of the expected returns on its asset classes, the highest return efficient portfolio can be 100% in stocks. Thus, the bonds to stocks ratio can be lower (higher) for aggressive (conservative) investors. Third, asset allocation decisions are examined in dynamic settings. Campbell and Viceira (2001) develop a dynamic model of optimal consumption and portfolio choice for infinite-lived investors in an economy with random real interest rates. They show that the notion of intertemporal hedging demand of Merton (1973) plays an important role in infinite-lived investors’ asset allocation decisions. The investors’ demands for long-term bonds can be increasing in risk aversion.1 The intuition underlying this direction is that bonds are more instruments of financing, instead of investing, since bonds are closer to risk-free than cash for long-term investors. 1 Similarly, under a different assumption of utility function, interest rate process, investment horizon, and/or consumption scheme, Brennan and Xia (2000), Sorensen (1999), and Wachter (2002) show that hedging demand has the potential to explain the asset allocation puzzle. 4
  5. 5. While the existing studies focus on the relationship between optimal risky allocations and risk aversion given a particular set of subjective return distributions at a particular time, little is known about professional investors’ asset allocation decisions. The main reason for this is that the existing studies have relied upon the assumption of an investor-manager. This usual convenience excludes the role of professional agents in making asset allocation decisions on behalf of individual investors. Since fund managers are agents and their objectives are likely to differ from those of investors in the funds, there is no obvious prediction from the existing theories about how mutual funds should optimally allocate their portfolios.2 In addition, a mutual fund is an aggregation of individual investors’ (sub-) portfolios. Through capital pooling, it is not clear whether the existing normative prescriptions applicable to individual investors can be directly mapped to mutual funds. To the best of our knowledge, there is no study examining how individual investors (through their agents) with different risk preferences actually rebalance their risky allocations over time when information sets shift. This study partially fills this gap by examining rebalancing decisions for pairs of asset allocation mutual funds that are managed by the same manager(s) but for different levels of risk-taking. This extension is important for several reasons. First, by examining fund managers’ actual asset allocation decisions, the study emphasizes the importance of the principal-agent relationship and capital pooling on portfolio choices.3 Second, unlike many prior studies that examined 2 In the literature of asset pricing, Goldman and Slezak (2003) model a delegated economy in which delegated fund management leads to rational mispicing similar to a bubble. It seems to be reasonable to question whether the rational portfolio choices in a delegated economy are different from those in a non- delegated economy. 3 One may argue that mutual funds’ asset allocation decisions are not suitable for testing portfolio choice models because these models assume one investor-manager. This paper focuses on the other side of the same argument. Our aim is to investigate whether these normative predictions can be still useful when the one investor-manager assumption is no longer applicable. After all, mutual funds are a $7 trillion industry 5
  6. 6. financial advisors’ recommendations, this study examines fund managers’ actual asset allocation decisions. Third, standard portfolio theories impose a restriction on the relative allocations to stocks between a risk seeker and a risk disliker. By focusing on rebalancing activities over time, this study is able to shed light on the scope of the usefulness of these models from a different perspective. Fourth, while the existing studies emphasize the economic contents of the bonds to stocks ratio whose theoretical relationship with risk aversion has been shown to be complex and ambiguous, the study focuses on the allocation to stocks whose relationship with risk aversion in the standard, static setting is rather straightforward.4 As a result, this study is able to provide cleaner tests. Finally, to further motivate our research design, consider the following scenario in asset allocation. Suppose that a mutual fund manager is in charge of two portfolios − one is aggressive and the other is conservative − and both portfolios have a market value of $100. Given the current information set provided to her, the manager allocates $50 and $30 to common stocks for the aggressive portfolio and the conservative portfolio, respectively. Suppose that one quarter later the manager becomes less bullish about stock markets and her portfolios are still worth $100 each. She decides to reduce the allocation to common stocks for the conservative portfolio from $30 to $24, i.e. a 20% reduction. Based on her updated subjective return distributions, what amount of common stocks will she allocate for the aggressive portfolio? Standard mean-variance models of portfolio choice suggest that it should be $40 with the same 20% reduction. The main contribution of the study is the establishment of a stylized fact that the rebalancing of common stocks serving more than 90 million investors. 4 In this study, the allocation to an asset class is normalized by the fund’s total market value so that the total allocations of a fund sum up to one. 6
  7. 7. in the aggressive (conservative) portfolio is disproportionately less (more) intense. To be specific, the manager is likely to reduce equity holdings for the aggressive portfolio by about 10% to around $45. The paper proceeds as follows. Section II introduces the hypothesis. Section III describes the data. Section IV presents empirical results. Section V generalizes the baseline results and provides robustness checks. Section VI investigates whether the documented results can be explained by dynamic models of portfolio choice. Section VII examines two plausible agency explanations. Concluding remarks are offered in the final section. II. Hypothesis We would have preferred to conduct our baseline investigation based on a general model of portfolio choice that explicitly considers the role of professional agents and capital pooling. However, because of the lack of theoretical understanding about portfolio choices in such a general setting, this study relies upon standard mean-variance models and their predictions. Although standard mean-variance models are valid under very restrictive conditions, they have their own strengths; they are widely taught and used, their predictions are straightforward, and they are amiable to testing. Based on this basic framework, this study considers a mutual fund manager who is able to invest in common stocks, bonds, and cash that is instantaneously risk-free. Given the information set available to the manager at time t = 0, standard one-period, mean-variance models of portfolio choice suggest that the vector of optimal risky asset allocations has the following form:  ws  1 -1  wb  = γ Σ π (1)   7
  8. 8. where ws is the optimal allocation to stocks, wb is the optimal allocation to bonds, γ is the parameter of constant relative risk aversion, Σ is the covariance matrix, and π is the 2×1 vector of risk premia for stocks and bonds. A fundamental property in equation (1) is that the demand for stocks is linearly decreasing in risk aversion. Furthermore, this property holds even when subjective return distributions shift over time. Therefore, equation (1) imposes a restriction on how rational agents with different degrees of risk aversion should rebalance their allocations to stocks over time. Specifically, suppose that a fund manager allocates proportions y and α×y of stocks to an aggressive portfolio with risk aversion γAGG at time t = 0 and 1, respectively. After controlling for information sets at time t = 0 and 1, equation (1) predicts that the fund manager would allocate proportions (γAGG/γCON)×y and (γAGG/ γCON)×α×y of stocks to a conservative portfolio with risk aversion γCON at time t = 0 and 1, respectively. This means that both the aggressive and the conservative portfolios would experience a rebalancing of stocks at the same magnitude of (α − 1) during this time period. In other words, although γCON and γAGG are unobservable, so long as the two portfolios’ risk preferences are well-defined and they are managed under the same information set the following identity would hold over the time period: %AGGs = %CONs (2) where %AGGs and %CONs are the percentage changes of the allocations to stocks for the aggressive portfolio and the conservative portfolio, respectively. Furthermore, if the allocation to stocks for the aggressive portfolio is higher than that for the conservative portfolio, i.e., AGGs > CONs, one would expect that for a population of asset allocation 8
  9. 9. decisions the standard deviation of AGGs is higher than that of CONs. Specifically, if AGGs = c×CONs where c > 1, equation (2) implies the following:5 std(AGGs) = c×std(CONs) > std(CONs) (3) Note that the restrictions in equations (2) and (3) would hold whenever asset allocation decisions are not financially constrained. This study focuses on investigating the restriction specified in equation (2).6 In reality, nevertheless, one would not expect the identity in equation (2) to hold perfectly because market frictions, such as self-imposed turnover and transaction cost constraints, illiquidity, and taxation considerations, may cause non-synchronous rebalancing between the paired portfolios. As a result, given a shift in subjective return distributions, Φ, the observable percentage changes of the allocations to stocks for the aggressive portfolio and the conservative portfolio, %aggs and %cons, may contain random errors: %aggs = f(Φ) + eagg (4a) %cons = f(Φ) + econ (4b) where both random errors eagg and econ are assumed to be distributed as N(0, σ).7 That is, the study assumes the sizes of random errors eagg and econ to be unknown, but they have the same value. This assumption appears to be reasonable because there is no obvious reason to believe that one of the paired portfolios is more influenced by non-synchronous rebalancing than the other. 5 Without loss of generality, one can write a time-series of the allocations to stocks for the conservative fund, {CONs,0 , CONs,1 ,…, CONs,N }, as {CONs,0 , a1× CONs,0 ,…, a1×…× aN× CONs,0}, where a1, …, aN are the sums of one and the percentage changes of the allocations to stocks for the fund at time 1, …, N, respectively. By equation (2), the allocations to stocks for the aggressive fund are {c×CONs,0 , c× a1× CONs,0 ,…, c× a1×…× aN× CONs,0}. 6 Equation (2) is amiable to inferring the economic significance of testing results. 7 The regression model in equation (5a) and (5b) is identified by making this extra assumption. Without it, the model is unidentified as the one in the classic demand and supply equations of macroeconomics. See Kendall and Stuart (1961) about the issue of identifiability. 9
  10. 10. Given this error structure, one would expect the slope estimate from one of the following two regressions on a sample of portfolio rebalances to be the reciprocal of the other regression:8 %aggs = α1 + β %cons + (eagg − econ) (5a) 1 %cons = α2 + %aggs + (econ − eagg) (5b) β Thus, testing the restriction in equation (2) amounts to testing whether α1 = 0, α2 = 0, β = 1, and 1/β = 1. 2. Data The study uses the Center for Research in Security Prices (CRSP) mutual fund database with data ending December 2002 to form pairs of aggressive and conservative mutual funds. The goal is to form a pair of asset allocation funds that are controlled for all informational attributes, except for the level of risk aversion. This study uses two screen tests to classify whether a fund is an asset allocation fund. The first test is to check the fund’s ICDI fund objective code.9 The fund passes the test if the fund is coded as “balanced,” “growth and income,” “income,” or “total return.”10 Because the ICDI fund objective code is not systematically available before 1993, the sample period begins in 1993. The second test is to check a fund’s actual allocations to stocks and bonds. Specifically, a fund is included for analysis only if its allocation to common stocks is no 8 Equations (4a) and (4b) imply that %aggs − eagg = %cons − econ. Rearranging the identity, we have equations (5a) and (5b). 9 The ICDI’s fund objective code identifies a fund’s investment strategy as identified by Standard & Poor’s Fund Services. 10 Therefore, a fund coded as “aggressive growth,” “high quality bonds,” “high-yield bonds,” “global bonds,” “global equity,” “Ginnie Mae funds,” “government securities,” “international equities,” “long-term growth,” “tax-free money market fund,” “government securities money market fund,” “high quality municipal bond fund,” “single-state municipal bond fund,” “taxable money market fund,” “high-yield money market fund,” “precious metals,” “sector funds,” “utility funds,” or “special funds” is excluded from analysis. 10
  11. 11. less than 10% of its market value and its aggregated allocation to corporate bonds, municipal bonds, and government bonds is no less than 10% of its market value. This test is to mitigate the possibility that one of the paired funds has an optimal corner solution in which investment decisions are financially constrained. This study uses two screen tests to control for informational homogeneity. The first screen requires both of the paired sample funds to be members of the same fund family. The second rule is that the paired sample funds must be managed by the same manager(s) and not by an investment committee whose constituent members are not identifiable. If within a fund family there are more than two qualified funds, we pick the pair that maximizes the difference in the allocations to common stocks. Once a fund pair is identified for a fund family, we track the pair and retrieve their portfolio weights through the end of 2002. The pair would exit the sample earlier if one of the following exclusion rules is met: (1) a fund is dead; (2) a fund becomes team managed; (3) the funds are managed by different managers;11 or (4) a fund’s aggregate allocation to preferred stocks, convertible bonds, warrants, and other unclassified securities is greater than 10% of its market value. The last rule is imposed because the hybrid and/or derivative nature of preferred stocks, convertible bonds, and warrants make them difficult to classify into equities or bonds. The number of the resulting sample pairs is 21. This seemingly low number is largely due to the fact that the majority of mutual funds specialize in security selection and many fund families do not offer more than one asset allocation fund. The list of the pairs is provided in Table 1. To classify one of the paired funds as aggressive and the 11 This rule ensures that even if the paired funds change the management company, they would be qualified only if they are still managed by the same manager(s). As a result, there would be no contamination in the data. 11
  12. 12. other as conservative, the study does not directly use their equity allocations as proxies. Instead, we check the prospectuses of the funds, search the fund family’s website, or contact a representative of the fund family to make the classification. However, for the sample fund pairs we find equity allocations to be perfect proxies for the degree of risk aversion in that self-described aggressive funds always have higher allocations to stocks than their conservative counterparts. Table 1 shows that the majority of the sample pairs either survive during the sample period or exit the sample for reasons that are not related to survivorship. There are three pairs that exit the sample due to merger or liquation: Prairie Funds: Managed Assets Fund/A is merged by an unknown merger; Kemper Horizon Fund: 20+ Portfolio/ A is merged by a bond fund whose subsequent allocation to stocks is zero; and State Street Research Strategy Portfolio: Moderate/S is liquidated. Consequently, we are unable to track their post-event asset allocations within the data selection framework. As a final resort, we retrieve the monthly returns on these merged or liquidated funds to check for abnormal return behavior preceding their exits. In Figure 1, a window of 12 months ending at the end of the funds’ sample periods is denoted by the end points of T-11 and T. Comparing these returns with the returns realized during their post-sample periods, it is evident that their post-sample returns are not unusually low, nor particularly volatile. One thus can infer that the asset allocation decisions of the three funds do not seem to be abnormal during the post-sample periods and that for this study survivorship bias does not appear to be a critical concern. 3. Empirical Results 12
  13. 13. The 21 sample fund pairs yield 104 pairs of portfolio weights and 83 pairs of asset allocation rebalances. On average, every fund pair has almost five sets of portfolio weights, allowing us to study four portfolio rebalances. The summary statistics of the 104 pairs of portfolio weights are reported in Table 2. As mentioned above, for our sample self-described aggressive funds always have higher allocations to stocks than their conservative counterparts. This is evident by the average allocation to stocks of 0.6516 for aggressive funds, which is almost twice that of 0.3417 for the conservative funds. According to equation (3), this feature would lead one to expect that the standard deviation of the allocations to stocks for aggressive funds is about twice that for conservative funds. However, the study finds the opposite. The standard deviation of the allocations to stocks for aggressive funds has a value of 0.1431 that is smaller than the standard deviation of the allocations to stocks of 0.1616 for conservative funds. The standard deviation to mean ratio of the allocations to stocks for aggressive funds (0.2196 = 0.1431/0.6516) is less than half of that for conservative funds (0.4700 = 0.1606/0.3417). This is in sharp contradiction to the predictions in equations (3). This observation clearly indicates that although conservative funds’ allocations to stocks are relatively low, their rebalances are relatively intense. On the other hand, the rebalances of the allocations to bonds and to cash in conservative funds are proportional to those in their aggressive counterparts, indicating that the documented intensity in the rebalances of the allocations to stocks among conservative funds is unique. Specifically, aggressive funds, on average, allocate 0.2594 of their market values to bonds with a standard deviation of 0.0999. The allocations to bonds for conservative funds have a mean value of 0.5280 and a standard deviation of 13
  14. 14. 0.1646. As a result, the standard deviation to mean ratio of the allocations to bonds for aggressive funds (0.3851 = 0.0999/0.2594) is not far away from that for conservative funds (0.3117 = 0.1646/0.5280). Similarly, conservative funds invest more in cash. The standard deviation to mean ratio of the allocations to cash for aggressive funds (1.1423 = 0.0875/0.0766) is close to that for conservative funds (1.0769 = 0.1218/0.1131). 12 Putting these results together, one can infer that the stylized disparity in the bonds to stocks ratios documented in Canner et al. (1997) is likely driven by the disparity in the allocation to stocks, but not by the disparity in the allocation to bonds. Given the prevailing emphasis on the bonds to stocks ratio in the existing literature, the study also reports this ratio. Table 2 shows that, consistent with Canner et al., conservative funds have a higher bonds to stocks ratio than aggressive funds. The mean values of the bonds to stocks ratio for aggressive and conservative funds are 0.4515 and 2.2287, respectively. In other words, the bonds to stocks ratio for a conservative fund is, on average, about four times that of an aggressive fund. Having characterized the sample, the study turns its attention to the restrictions in equations (2). That is, the percentage changes of allocations to stocks for aggressive funds need to be equal to those for conservative funds if fund rebalancing follows standard mean-variance models. To test these restrictions, one may apply ordinary least squares (OLS) to the regression specifications in equations (5a) and (5b). But this is statistically problematic. As shown in equations (4a) and (4b), both %aggs and %cons are endogenous and are functions of the shifts in subjective return distributions, subject to 12 Partially due to the sample selection criteria that impose an upper bound on the aggregate allocation to preferred stocks, convertible bonds, warrants, and other unclassified securities, the sample fund pairs have less than an average of 1% allocation to preferred stocks, convertible bonds, warrants, or other unclassified securities. Nevertheless, it is interesting to note that conservative funds allocate more into preferred stocks than their aggressive counterparts (0.0073 vs. 0.0038). Although the result is beyond the scope of the paper, fund mangers seem to view preferred stocks more like bonds than common shares. 14
  15. 15. random errors. It is well-known that the use of OLS assumes that the regressor(s) are fixed and exogenous while the dependent variable is allowed for random errors. Because of the violation of this underlying assumption, the use of OLS results in non-compatible estimates for β and 1/β in that they are not the reciprocal of the other. One may also apply a t-test for two population means on the absolute values of the percentage changes of allocations to stocks. Absolute values are used here because more intense rebalancing means high (low) percentage changes of allocations to stocks when they are positive (negative), and vice versa. This test leads to a t-statistic of 4.45 that is statistically significant at the 1% level. However, this simpler test is appropriate only when the populations are normally distributed. This is obviously not the case for the two populations of the absolute values of the percentage changes of allocations to stocks because they are bounded by zero. The study proposes a symmetrical fitting method, called total least squares (TLS), to test the restrictions in equation (2). The method of TLS fits a linear manifold of dimensional n to a given set of data points in ℜn+1 such that the sum of orthogonal squared distances from these points to the manifold reaches a minimum. By doing so, the method simultaneously minimizes the random errors associated with %aggs and %cons. As a result, the symmetrical structure of TLS is able to ensure that the slope estimates from regression specifications (5a) and (5b) are the reciprocal of the other. This method of fitting has a long history in the literature of statistics where the method is known as orthogonal regression (Adcock (1878)).13 A numerical stable algorithm of TLS is provided in the Appendix. 13 This type of fitting is not scale invariant. This property, along with data scaling, is sometimes utilized for model identification. For this study, data scaling is not needed because random errors eagg and econ are assumed to be of the same size. See Litzenberger and Ramaswamy (1979) for the use of data scaling. 15
  16. 16. The TLS regression results are reported in Table 3. Panel A reports the regression results based on equation (5a). On average, the rebalances of stocks in aggressive funds are about 40% of those in conservative funds. The estimate for β has a value of 0.4010. This value is considerably lower than the hypothesized value of one. Based on equation (5b), Panel B shows that, with the use of TLS, reversing the regression specification has no impact on statistical inferences. The estimate of 2.4940 for 1/β is exactly the reciprocal of the estimate for β (2.4940 = 1/0.4010). Consequently, consistent with the results documented in Panel A, this regression shows that the magnitudes of the rebalances of stocks in conservative funds are, on average, about 2.5 times those of aggressive funds.14 To assign statistical significance to TLS estimates, the study randomly resamples the initial sample of asset allocation rebalances, with replacement, to generate a large number of bootstrapped samples. The number of simulation repetitions used is 10,000. We then apply TLS to each of these bootstrapped samples to obtain the empirical distribution of TLS estimates. Once the distribution is obtained, the calculation of the bootstrapped p-values and statistical inferences can be made in the usual way. The use of bootstrapped tests is beneficial because the distribution of the population of the percentage changes of the allocations to stocks is, by construction, non-normal.15 These percentage changes are bounded by -100% when mutual funds do not short sell, which is the usual practice in this industry. As a result, our non-parametric testing framework should be more robust with respect to the unknown distribution of the population. 14 Taking the intercept term into consideration, a 10% decrease (increase) in the allocation to stocks for aggressive funds is approximately associated with a 20% decrease (increase) in the allocation to stocks for conservative funds. 15 We check our sample percentage changes of the allocations to stocks. They exhibit x2–like distributional characteristics. 16
  17. 17. In Table 3 the bootstrapped p-values are contained in parentheses. The p-values for the intercept terms of the two TLS regressions, 0.5324 and 0.5358, show that the two intercept terms are not statistically different from zero at any conventional level. For the two slope estimates, the study computes their p-values against the hypothesized value of one under the null of standard mean-variance models. The p-values of 0.0000 and 0.0038 for the two slope estimates indicate that β and 1/β are statistically different from one in which β (1/β) is statistically lower (higher) than one. The evidence is clearly at odds with the optimality restrictions in equation (2). 4. Generalization and Robustness Checks The results documented thus far suggest that the optimality restrictions in equation (2) are violated. Are the results robust? For one thing, although pairing asset allocation funds with the same manager(s) enables clean, experimental tests, one may argue that the study does not make a strong case because of the small sample size employed. To address this concern and provide a larger sample size for the subsequent robustness checks,16 the study expands the original dataset to include those fund pairs that belong to the same fund families, but are managed by different managers. This expansion increases the number of sample fund families from 21 to 50 and the number of asset allocation rebalance pairs from 83 to 231. The TLS regression results for the augmented dataset are reported in Table 4. The results are stronger than those presented in the baseline. On average, the rebalances of stocks in aggressive (conservative) funds are about 28% (3.5 times) of those in conservative (aggressive) funds. The bootstrapped p-values for β and 1/β are both less 16 Most of the subsequent robustness checks involve partitioning the dataset. For practical reasons, the baseline dataset is, sometimes, too small to be further partitioned. Moreover, the bootstrap estimator is a consistent estimator. The bootstrap test works better when the sample size is adequately large. 17
  18. 18. than 0.0001. It appears that the baseline results can be generalized to a broader setting in which the purity of informational homogeneity is lower. One may also wonder whether the sample portfolio rebalances are truly free from financial constraints. It is well-known that mutual funds usually do not use financial leverage. Consequently, when portfolio optimization is financially constrained, the restrictions in equation (2) would not be valid. To mitigate the concern, the study excluded those funds whose allocations to stocks are less than 10% of their market values and their aggregated allocations to corporate bonds, municipal bonds, and government bonds are less than 10% of their market values. By doing so, the baseline results were free from corner solutions in which the allocation to stocks or bonds is 100%. However, the baseline did not address the possibility that the optimal aggregate allocation to stocks and bonds may reach 100%. How would this possibility affect the results when it is realized? To check this, we exclude those portfolio weight pairs whose allocations to cash are not both greater than 3%. This exclusion rule reduces the sample rebalance pairs from 231 to 81 for the augmented dataset. The empirical results of TLS regressions on the 81 portfolio rebalances are reported in Panels A and B of Table 4. It is evident that the estimates for β and 1/β are still far away from one. The two estimates are 0.3818 and 2.6195, respectively. Because of a smaller sample size, the bootstrapped p-values for β and 1/β increase to 0.0002 and 0.0094, respectively. Nevertheless, they are still statistically significant at the 1% level, indicating that the results are robust. 17 Finally, the stylized fact of greater absolute percentage changes of allocations to stocks for conservative funds than for aggressive funds seems to be consistent with a 17 In addition to 3%, the study also uses 5% as the exclusion rule. The slope estimates of 0.3780 and 2.6458 for β and 1/ β are almost identical to those with the use of 3% exclusion rule. The statistical significance drops to the 5% level due to a smaller sample size of 56. 18
  19. 19. transaction costs story. As shown above, conservative funds invest more in bonds. It is well-known that trading Treasury bonds involves little transaction costs. Therefore, it may be more cost-efficient for rebalancing in conservative funds. To check this possibility, the study partitions the augmented rebalance pairs into four subsamples according to whether the average allocation of bonds in Treasury bonds at the beginning of the rebalance period is no greater than 25%, greater than 25% and no greater than 50%, greater than 50% and no greater than 75%, or greater than 75%.18 The four subsamples have sample sizes of 26, 71, 64, and 70, respectively. The idea underlying this partition is that if the documented stylized fact is due to differential transaction costs, one would expect the relative aggression in conservative funds to be more profound in those rebalance pairs with a high ratio of Treasury bond allocation to bond allocation. However, the TLS regression results in Panels C, D, E, and F of Table 4 show that regardless of the degree of concentration in Treasury bonds, the estimates for 1/β are all greater than one.19 Specifically, they have values of 2.0937, 2.0340, 6.3574, and 2.8040, respectively. That is, even among those rebalance pairs whose bond holdings are mostly in corporate bonds and/or municipal bonds, the documented relative aggression in conservative funds persists. Because of smaller sample sizes, the four subsamples yield a wide range of bootstrapped p-values for 1/β. They are 0.0702, 0.0174, 0.1506, and 0.0190, respectively. 5. Dynamic Models 18 The study experiments with other cutoffs. The study also uses aggressive funds’ allocations of bonds in Treasury bonds and conservative funds’ allocations of bonds in Treasury bonds to form subsamples. The reported results are insensitive to these changes. 19 For a more concise presentation, the study only reports the regression results with the use of %aggs as the regressor for this and the following analyses. Because of the symmetrical nature of TLS, the use of %cons as the regressor reaches the same conclusion. 19
  20. 20. When considering a multi-period setting in which investment opportunities are time varying, dynamic models of portfolio choice often lead to opposite predictions.20 The main reason for this is that the relative importance between myopic demand and hedging demand varies both endogenously and exogenously. Hedging demand is a complex function of many factors, such as risk aversion, return predictability, and time horizon. Ideally, one should use the dynamic counterpart of equation (1) to test the rationality of fund reallocations. Unfortunately at this point realistically complex demand functions do not exist because they are difficult to solve analytically. The available demand functions are mostly based on strong assumptions, such as constant risk premium, that would not serve the purpose of this study.21 As a result, the study cannot offer a definitive answer as to whether differential hedging demand is responsible for the documented stylized fact.22 Yet there is empirical evidence suggesting that differential hedging demand cannot fully explain our results. To see this, note that the explanation based on differential hedging demand is plausible only if conservative funds have a higher hedging demand for stocks than aggressive funds. The calibration results of Campbell, Chan, and Viceira (2003) and Campbell and Viceira (2002, p. 111) show the opposite. According to these authors’ VAR estimates, hedging demand for stocks is a hump-shaped function of risk aversion, which peaks when total demand for stocks is above 100%; that is, 20 Ingersoll (1987, p.245) provides an example where aggressive investors have lower demand for risky assets than conservative investors. Liu (2001) shows that holdings of risky assets might increase with risk aversion even when risk premiums are strictly positive. 21 Examples of these analytical demand functions can be found in Brennan and Xia (2000, eq. (15)) and Sorensen (1999, eq. (7)). 22 In fact, Campbell and Viceira (1999) and Campbell, Chan, and Viceira (2003) suggest that aggressive portfolios should move in and out of stocks more aggressively than conservative portfolios. 20
  21. 21. conservative investors actually have a lower hedging demand for stocks than aggressive investors so long as they do not use financial leverage.23 Furthermore, even if conservative funds do have a higher hedging demand for stocks than aggressive funds, hedging demand needs to be quite volatile such that differential hedging demand is able to drive the stylized fact. Again, Campbell, Chan, and Viceira show that hedging demand is much more stable than myopic demand because hedging demand can change sign only when investors take a short position in stocks (Kim and Omberg (1996), Campbell and Viceira (1999)). Finally, differential hedging demand is a more useful explanation if conservative funds are tiled to investors with a longer time horizon than aggressive funds do.24 To check this, we employ the CRSP mutual fund database to identify mutual fund families and use the Yahoo search engine to locate their websites. Among those fund families which provide investment planning services on their websites, none suggest an investor with a longer time horizon to invest in a relatively conservative fund.25 6. Agency Issues The results documented in the previous section are disturbing. If professional asset allocation decisions are indeed suboptimal, what might explain this? A natural suspect is that mutual funds are run by managers whose objectives may differ from those of the investors in the funds. As a result, these asset allocation decisions may be deemed rational from managers’ perspectives, but not from investors’ perspectives. 23 Campbell, Chan, and Viceira (2003) find that bonds are more useful for hedging conservative investors’ intertemporal risk. 24 Campbell and Viceira (1999) and Campbell, Chan, and Viceira (2003) show that multi-period investors should rebalance their equity holdings more aggressively than one-period, myopic investors. 25 An example of Online investment planners can be found at http://askmerill.ml.com/example/display/1,,534,00.pdf. 21
  22. 22. Arguably, there exist endless types of agency problems in the mutual fund industry, ranging from illegal late trading to minimizing efforts. For expositional purposes, this study focuses on two aspects of agency issues. First, the competition in the mutual fund industry is often characterized by gaming behavior. For instance, high-risk strategy and fee waiving are used to improve tournament outcomes in the second half of the annual performance evaluation cycle (Brown, Harlow, and Starks (1996), Chevalier and Ellison (1997), Christoffersen (2001)).26 If the relative aggression of conservative funds is related to tournament effects, one would expect the stylized fact to be more pronounced in the second half of the year. The study uses the augmented dataset to form two subsamples that consist of those rebalance pairs whose time spans are within the first half year and the second half year, respectively.27 This procedure results in the first-half and second-half subsamples of 48 and 38 rebalance pairs, respectively.28 The TLS regression results of the two subsamples are reported in Panels A and B of Table 5. The results suggest that gaming behavior does not seem to be a likely cause of the documented stylized fact. The two estimates for 1/β, 5.2717 and 5.6493, are both greater than one. Again, due to small sample sizes their statistical significances differ greatly. The bootstrapped p-values for the two subsamples are 0.3298 and 0.0034, respectively. Second, given the well-documented notion of heuristics in decision making (Tversky and Kahneman (1974), Read and Loewenstein (1995), Benartzi and Thaler (2001)), it is plausible that the seeming suboptimality is related to fund managers’ 26 In contrast, Busse (2001) finds that likely winners increase fund riskiness more than likely losers. 27 For example, a rebalance pair between 12/1999-06/2000 is grouped into the first-half subsample; a rebalance pair between 09/2001-12/2001 is grouped into the second-half subsample; and a rebalance pair between 03/2000-09/2000 is excluded from this analysis. 28 Before 2000, CRSP mutual fund database contains mostly annual rebalances. These rebalances are excluded from this analysis because they span over the first half year and the second half year. 22
  23. 23. intuitive trading behavior. Because of the prevailing emphasis on the bonds to stocks ratio in the existing theoretical and empirical studies, the study conjectures that if a heuristic does exist, it is most likely to be based on the bonds to stocks ratio. To investigate the possibility of heuristic rebalances, the study replaces the metric of the percentage changes of the allocations to stocks by the metric of the percentage changes of the bonds to stocks ratio. The TLS regression results based on the latter metric with the use of the augmented dataset are reported in Panel C of Table 5. The correspondence between the percentage changes of the bonds to stocks ratio for aggressive funds and the percentage changes of the bonds to stocks ratio for conservative funds is close to, but not quite, one-to-one. The estimate for 1/β is reduced from 3.5376 to 1.6083. Nevertheless, the corresponding bootstrapped p-value increases to 0.6786; no longer statistically significant at any conventional level. Thus, there is weak evidence suggesting that fund managers may heuristically relate the bonds to stocks ratios between paired funds. That is, if they increase (decrease) the bonds to stocks ratio for a relatively aggressive fund, they are likely to increase (decrease) the same ratio at the same magnitude for a relatively conservative fund as well, and vice versa. 7. Conclusions The study employs standard, static models of portfolio choice and tests their optimality restrictions on the relative allocations to common stocks between a relatively aggressive fund and a relatively conservative fund over time. These restrictions are found to be violated for a sample of asset allocation mutual funds. Specifically, rebalancing of stocks in aggressive (conservative) funds is disproportionally too light (intense). The study acknowledges that standard models of portfolio choice were not 23
  24. 24. specifically developed for explaining mutual funds’ asset allocation decisions because this set of models are valid under restrictive conditions; these models are used in this study because they are amiable to testing and there seems no feasible alternative at this point. The study also considers dynamic models of portfolio choice and their diverse predictions. However, the existing empirical evidence from the literature of dynamic portfolio choice does not help resolve the puzzle. Consequently, the rationality of professional asset allocation decisions is called into question. Admittedly, the study is empirical and expositional in nature. As a result, the search conducted in the study for finding the cause of the documented stylized fact is incomplete. For instance, although the study finds weak evidence that fund managers may heuristically relate the bonds to stocks ratios between funds with different degrees of risk aversion, the study comes short of presenting a strong case and establishing causality. Thus, the source of the heuristic needs further investigation. In addition, one should not underestimate the possibility that future theoretical extension on rational portfolio choice, either from investors’ or managers’ perspectives, is able to explain the documented stylized fact. In fact, we believe that this research direction will be fruitful and, if so, this would be our most important contribution because the documented stylized fact warrants more theoretical developments to help us better understand professional portfolio choices. 24
  25. 25. References Adcock, R.J., 1878, “A Problem in Least Squares,” The Analyst, 5, 53-54. Benartzi, S., and R.H. Thaler, 2001, “Naïve Diversification Strategies in Defined Contribution Saving Plans,” American Economic Review, 91, 79-98. Brennan, M.J., and Y. Xia, 2000, “Stochastic interest rates and the bond-stock mix. European Finance Review 4, 197-210. Brinson, G.P., L.R. Hood, and G.P. Beebower, 1991, “Determinants of Portfolio Performance II: An Update,” Financial Analysts Journal, 47, 40-48. Brown, K.C., W.V. Harlow, and L.T. Starks, 1996, “Of Tournaments and Temptations: An Analysis of Managerial Incentives in the Mutual Fund Industry,” Journal of Finance, 51, 85-110. Campbell, J.Y., Y.L. Chan, and L.M. Viceira, 2003, “A Multivariate Model of Strategic Asset Allocation,” Journal of Financial Economics, 67, 41-80. Campbell, J.Y., and L.M. Viceira, 1999, “Consumption and Portfolio Decisions When Expected Returns Are Time Varying,” Quarterly Journal of Economics, 114, 433-495. Campbell, J.Y., and L.M. Viceira, 2001, “Who Should Buy Long-Term Bonds?” American Economic Review, 91, 99-127. Campbell, J.Y., and L.M. Viceira, 2002, Strategic Asset Allocation: Portfolio Choice for Long-Term Investors, Oxford University Press, Oxford. Canner, N., N.G. Mankiw, and D.N. Weil, 1997, “An Asset Allocation Puzzle,” American Economic Review, 87, 181-191. Chevalier, J., and G. Ellison, 1997, “Risk Taking by Mutual Funds as A Response to Incentives,” Journal of Political Economy, 105, 1167-1200. Christoffersen, S.E.K., 2001, “Why Do Money Fund Managers Voluntarily Waive Their Fees?” Journal of Finance, 56, 1117-1140. Elton, E.J., and M.J. Gruber, 2000, “The Rationality of Asset Allocation Recommendations,” Journal of Financial and Quantitative Analysis, 35, 27-41. Golub, G.H., and C.F. Van Loan, 1980, “An Analysis of the Total Least Squares Problem,” SIAM Journal of Numerical Analysis, 17, 883-893. Ingersoll, J.E., 1987, Theory of Financial Decision Making, Rowman and Littlefield, Maryland. 25
  26. 26. Kendall, M.G, and A. Stuart, 1961, The Advanced Theory of Statistics, Vol. 2, Charles Griffin & Company Limited, London. Kim, T.S., and E. Omberg, 1996, “Dynamic Nonmyopic Portfolio Behavior,” Review of Financial Studies, 9, 141-161. Litzenberger, R.H., and K. Ramaswamy, 1979, “The Effect of Personal Taxes and Dividends on Capital Asset Prices,” Journal of Financial Economics, 7, 163-195. Liu, J., 2001, “Dynamic Portfolio Choice and Risk Aversion,” working paper, UCLA. Markowitz, H.M., 1952, “Portfolio Selection,” Journal of Finance, 7, 77-91. Merton, R.C., 1973, “An Intertemporal Capital Asset Pricing Model,” Econometrica, 41, 867-887. Read, D., and G. Loewenstein, 1995, “Diversification Bias: Explaining the Discrepancy in Variety Seeking between Combined and Separated Choices,” Journal of Experimental Psychology: Applied, 1, 34-49. Shefrin, H., and M. Statman, 2000, “Behavioral Portfolio Theory,” Journal of Financial and Quantitative Analysis, 35, 127-151. Sorensen, C., 1999, “Dynamic Asset Allocation and Fixed Income Management,” Journal of Financial and Quantitative Analysis, 34, 513-531. Trefethen, L.N., and D. Bau, 1997, Numerical Linear Algebra, SIAM, Philadelphia. Tversky, A., and D. Kahneman, 1974, “Judgment under Uncertainty: Heuristics and Biases,” Science, 185, 1124-1131. Van Huffel, S., and J. Vandewalle, 1989, “Analysis and Properties of the Generalized Total Least Squares Problem A X ≈ B When Some or All Columns of A Are Subject to Errors,” SIAM Journal of Matrix Analysis and Application, 10, 294-315. Wachter, J.A., 2002, “Risk Aversion and Allocation to Long-Term Bonds,” working paper, NYU. 26
  27. 27. Table 1 Fund pairs Initial Fund Name (ICDI_NO) Sample Period Exit Code Prairie Funds: Managed Assets Fund/A (01693) 199512-199606 M Prairie Funds: Managed Assets Income Fund/B (00189) Phoenix Total Return Fund/A (00200) 199509-199603 D Phoenix Series Fd: Balanced Fund/A (20260) Alliance Growth Investors Fund/A (00724) 199909-200202 A Alliance Conservative Investors Fund/A (00718) Fidelity Asset Manager: Growth (00816) 199406-200203 PG Fidelity Asset Manager: Income (03774) T. Rowe Price Personal Strategy Fds: Growth (12398) 199509-200209 A T. Rowe Price Personal Strategy Fds: Income (00971) Primary Trend: Trend Fund (00990) 199909-200006 PL Primary Trend: Income Fund (26260) J Hancock Sovereign Investors Fund/A (01150) 199612-200006 D J Hancock Sovereign Balanced Fund/A (01614) Dreyfus Asset Allocation Fd: Total Return Port (04502) 199509-199612 D Dreyfus Balanced Fund (01674) Morgan Stanley Dean Witter Balanced Growth/C (01709) 199912-200212 A Morgan Stanley Dean Witter Balanced Income/B (08927) Kemper Horizon Fund: 20+ Portfolio/A (02939) 199603-200006 M Kemper Horizon Fund: 10+ Portfolio/A (02945) Pacific Advisors: Balanced Fund/A (03248) 199612-200109 PG Pacific Advisors: Income and Equity Fund/A (03254) CitiSelect Folio 400 Series (04947) 199912-200006 A CitiSelect Folio 200 Series (04943) State Street Research: Managed Assets Fund/A (20650) 199603-199812 L State Street Research Strat Port: Moderate/S (05242) BT Investment Lifecycle Funds: Long Range Fund (05628) 199406-200209 A BT Investment Lifecycle Funds: Short Range Fd (05624) Putnam Asset Allocation: Balanced Port/A (07096) 199603-199612 PG Putnam Asset Allocation: Conservative Port/A (07102) Federated: Mngd Growth/Sel (08888) 199406-199812 PG Federated: Mngd Growth & Income/Sel (08886) INVESCO Industrial Income Fund (08930) 199612-199806 D INVESCO Multiple Asset Fds: Balanced Fund (09412) Merrill Lynch Asset Growth/A (11268) 199509-199909 T Merrill Lynch Asset Income/A (11256) Avesta Trust: Risk Manager-Balanced Fund (11638) 199506-199609 PG Avesta Trust: Risk Manager-Income Fund (11636) Aetna Generation Funds: Crossroads/C (13279) 199812-200109 T Aetna Generation Funds: Legacy/C (13281) USAA Invest Tr: Growth & Tax Strategy Fund (20940) 199303-199612 D USAA Mutual Fund: Income Fund (24290) This table lists 21 pairs of mutual funds used in the study. The fund in the first row of each pair is more aggressive than the fund in the second row. The sample pairs that are still alive as of December 2002 are coded as A (alive). The pairs that exit the sample earlier are coded as D (managed by different managers), L (liquidated), M (merged), PG (aggregated percentage invested in preferred stocks, convertible bonds, warrants, and other unclassified securities is greater than 10%), PL (percentage invested in common shares is less than 10% or the aggregated percentage invested in corporate bonds, municipal bonds, and government bonds is less than 10%), or T (team managed). 27
  28. 28. Table 2 Summary statistics Aggressive Conservative Mean Std Max Min Mean Std Max Min Common Stocks 0.6516 0.1431 0.8740 0.3360 0.3417 0.1606 0.6869 0.1027 Bonds 0.2594 0.0999 0.5200 0.0850 0.5280 0.1646 0.8700 0.2067 Cash 0.0766 0.0875 0.4392 -0.0269 0.1131 0.1218 0.5320 0.0000 Preferred Stocks 0.0038 0.0097 0.0478 0.0000 0.0073 0.0198 0.0967 0.0000 Convertible Bonds 0.0005 0.0023 0.0190 0.0000 0.0007 0.0036 0.0286 0.0000 Warrants 0.0000 0.0002 0.0011 0.0000 0.0000 0.0000 0.0001 0.0000 Other 0.0081 0.0193 0.0986 -0.0142 0.0092 0.0218 0.0962 -0.0447 Bond/Stock Ratios 0.4515 0.2764 1.1818 0.1195 2.2287 1.8343 8.1795 0.3009 This table reports summary statistics of the sample. The reported numbers are based on the portfolio weights allocated to common shares, bonds (aggregate of corporate bonds, municipal bonds, and government bonds), cash, preferred stocks, convertible bonds, warrants, and other unclassified securities. The sample size is 104. 28
  29. 29. Table 3 TLS regression results Intercept Slope Panel A: %aggs =α1 + β %cons + (eagg − econ) (N=83) 0.0078 0.4010 (0.5324) (<0.0001)** Panel B: %cons =α2 + 1/β %aggs + (econ − eagg) (N=83) -0.0194 2.4940 (0.5358) (0.0038)** Panel C: %aggs =α1 + β %cons + (eagg − econ) (N=231) 0.0045 0.2827 (0.4850) (<0.0001)** Panel D: %cons =α2 + 1/β %aggs + (econ − eagg) (N=231) -0.0161 3.5376 (0.4846) (<0.0001)** This table reports TLS regression results. %aggs and %cons are the percentage changes of the allocations to common stocks for aggressive and conservative mutual funds, respectively. Bootstrapped p-values are contained in parentheses. ** Significant at the 1% level. 29
  30. 30. Table 4 Robustness checks Intercept Slope Panel A: %aggs =α1 + β %cons + (eagg − econ) (%Cash > 3%) 0.0006 0.3818 (0.9848) (0.0002)** Panel B: %cons =α2 + 1/β %aggs + (econ − eagg) (%Cash > 3%) -0.0014 2.6195 (0.9878) (0.0094)** %T - Bonds Panel C: %cons =α2 + 1/β %aggs + (econ − eagg) ( ≤ 25%) %Bonds -0.0333 2.0937 (0.2808) (0.0702) %T - Bonds Panel D: %cons =α2 + 1/β %aggs + (econ − eagg) (25% < ≤ 50%) %Bonds -0.0028 2.0340 (0.8932) (0.0174)* %T - Bonds Panel E: %cons =α2 + 1/β %aggs + (econ − eagg) (50% < ≤ 75%) %Bonds 0.0002 6.3574 (0.9268) (0.1506) %T - Bonds Panel F: %cons =α2 + 1/β %aggs + (econ − eagg) (75% < ) %Bonds -0.0194 2.8040 (0.7284) (0.0190)* This table reports TLS regression results. %aggs and %cons are the percentage changes of the allocations to common stocks for aggressive and conservative mutual funds, respectively. The sample size for the regressions in Panels A and B is 81. The sample size for the regressions in Panels C, D, E, and F are 26, 71, 64, and 70, respectively. Bootstrapped p-values are contained in parentheses. * Significant at the 5% level. ** Significant at the 1% level. 30
  31. 31. Table 5 Agency issues Intercept Slope Panel A: %cons =α2 + 1/β %aggs + (econ − eagg) (First Half) 0.0179 5.2517 (0.9696) (0.3298) Panel B: %cons =α2 + 1/β %aggs + (econ − eagg) (Second Half) -0.0751 5.6493 (0.0968) (0.0034)** conb aggb Panel C: % =α2 + 1/β % + (econ − eagg) cons aggs 0.0051 1.6083 (0.2166) (0.6786) This table reports TLS regression results. %aggs and %cons are the percentage changes of the allocations to common stocks for aggressive and conservative mutual funds, respectively. % aggb /aggs and %conb/cons are the percentage changes of the bonds to common stocks ratio for aggressive and conservative mutual funds, respectively. The sample sizes for the regressions in Panels A and B are 48 and 38, respectively. The sample size for the regression in Panel C is 231. Bootstrapped p-values are contained in parentheses. * Significant at the 10% level. 31
  32. 32. Figure 1 Monthly returns on three dead funds T denotes the end of the sample period. 32
  33. 33. Appendix The TLS estimator can be applied to a regression model that includes both exactly known regressors and those measured with errors. Consider a multivariate relationship Y = X β, where Y is one-dimensional and is m × 1, X is m × n, β is n × 1. One can think of X as an aggregate of a matrix of some known variables and a matrix of unobservable variables. That is, the study partitions X as [X1 , X2] in which the exactly known X1 is m × n1 and may include an intercept term, X2 is unobservable and is m × n2, and n1 + n2 = n. The corresponding parameters β1 and β2 are n1 × 1 and n2 × 1, respectively. Both X2 and Y can only be observed with random errors such that the observed X2Σ = X2 + Σ2 and the observed Y∆ = Y + ∆, where Σ2 and ∆ are error matrices. The study defines the observed XΣ as [X1 , X2Σ], and assumes that XΣ is of full rank. Suppose that the observations are properly scaled to achieve model identification, i.e., the columns in [ Σ2, ∆] have the same variance. Then, the TLS problem seeks to: Minimize || [X2Σ , Y∆] − [X2ΣTLS , Y∆TLS] || F [X2ΣTLS , Y∆TLS] ∈ ℜm × (n2 + 1) subject to range(Y∆TLS) ⊆ range (XΣTLS) = range ([X1 , X2ΣTLS]) where F denotes Frobenius norm. The Frobenius norm of an m × n matrix P is defined as || P ||F = (trace(PT P))1/2. Once the minimizing X2ΣTLS and Y∆TLS are found, then any β satisfying  b1  [X1 , X2ΣTLS]   ≡ [X1 , X2ΣTLS] βTLS = Y∆TLS  β 2TLS  33
  34. 34. is called the TLS solution, βTLS. Computationally, the solution involves computing a QR factorization of an augmented matrix, 29 solving the TLS problem of reduced dimension, and using the TLS solution to deduce the OLS solution for the exactly known regressors. That is, the study first performs n1 Householder triangularization to compute QR factorization of the augmented matrix [X1 , X2Σ , Y∆] such that:  R11 R12 R1Y  QT [X1 , X2Σ , Y∆] =    0 R 22 R 2Y  where Q is unitary, i.e., QT Q = I, R11 is a n1 × n1 upper triangular matrix, R12 is n1 × n2, R22 is (m − n1) × n2, R1Y is n1 × 1, and R2Y is (m − n1) × 1. This transformation separates the OLS and TLS components of the above problem and makes computation tractable. The next step is to compute the reduced TLS solution β2TLS of R22 β ≅ R2Y. This involves the SVD of [R22 , R2Y] [Golub and Van Loan (1980)]: [R22 , R2Y] = U Σ VT Σ = diag(τ1, ..., τn2+1) V = [v1, ....., vn2+1] where τ1, ..., τn2+1 are singular values, and U and V are matrices with orthonormal columns. The reduced TLS solution β2TLS of R22 β ≅ R2Y exists and is unique [Van Huffel and Vandewalle (1989)]: β2TLS = −1/vn2+1, n2+1 [v1, n2+1, ....., vn2, n2+1] T 29 See Trefethen and Bau (1997, p. 41) for details about QR factorization. 34
  35. 35. To find the n1 × 1 OLS solution b1, the study then projects the reduced vector [Y∆ − X2Σ β2TLS] into range(X1) by solving R11 b1 = R1Y − R12 β2TLS. 35

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