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Abstract
The authors explain optimized portfolios' poor out-of-sample performance (to minimize tracking error relative to a given benchmark, while achieving a specified expected excess return) in the presence of estimation error in the underlying asset means and covariances. The theoretical bias adjustments for this estimation risk developed by the authors involves taking mathematical expectations of asymptotically expanded future returns of portfolios formed with estimated weights. They provide closed-form adjustments for estimates of the expectation and standard deviation of the portfolio's excess returns. The adjustments significantly reduce bias in global equity portfolios, reduce the costs of rebalancing portfolios, and are robust to sample size and non-normality. By using these approximation methods before investing, it may be possible to assess the effect of statistical estimation error on tracking-error-optimized portfolio performance.
Full text
(ProQuest: ... denotes formulae omitted.)
The performance of a fund relative to a chosen benchmark has traditionally been a key determinant of the fund's attractiveness as an investment. Hence, a commonly implemented strategy throughout the investment community is to minimize the tracking error relative to a given benchmark, subject to a given expected level of positive excess return.
In spite of their wide use, expected excess returns and tracking errors have been difficult to estimate, especially in mean-variance optimized portfolios, in part due to the statistical noise contained in the means and variances of asset returns estimated from historical data (see Scherer and Martin [2007] for a review). Chan et al. [1999] look at forecasting the second moments of the returns in order to reduce the tracking error, while Pachamanova [2006] suggests using robust optimization.
Our research looks at the effect of estimation error on the expected excess return and tracking error for portfolios optimized using mean-variance optimization. We derive asymptotic formulae that eliminate dominant terms of the bias arising from noisy estimates, hence improving the implementation power of the tracking measures. We extend to the case of tracking-erroroptimized portfolios the methods developed by Siegel and Woodgate [2007] for meanvariance optimized portfolios. We use the method of statistical differentials to find Taylor-series approximations to expectations of random variables, obtaining theoretical results that are asymptotically correct when the number of time periods is large and that remain statistically consistent when estimated values are substituted for unknown parameters. Our approach is non-Bayesian, not relying upon prior assumptions on the unknown parameters.
We perform the following thought experiment. Suppose an investor forms classical sample estimates of asset means, variances, and covariances and uses them as if they were the true assets' distribution parameters to form a tracking-error-optimized portfolio that achieves a specified expected excess return with respect to a given benchmark. We refer to these classical sample performance estimates as "naïve" estimates, since they do not take into account the estimation error stemming from using a sample rather than the population in determining these parameters. If the estimation error wrongly suggests that an asset will have a high expected return, then an optimized portfolio heavily invested in this asset will be disappointing. In particular, estimates of the naïve portfolio's expected excess return tend to be biased upwards, while estimates of the naïve portfolio's tracking-error tend to be biased downwards, resulting in nominally tracking-error-optimized portfolios that are over-optimistic, in the sense that such an investor will believe that she can achieve a higher expected excess return and lower tracking error than is actually available from the performance of her portfolio. We quantify this overoptimism bias in closed-form asymptotic formulae, which help in two ways when statistical noise influences portfolio weights. First, there may be a systematic component that moves the portfolio's expected excess return away from its target; this effect is captured by our expected excess return adjustment. Second, to the extent that the portfolio weights reflect the variability of the noise and the estimated variability, the variance of the portfolio excess returns will increase; this effect is picked up by our adjustment for the tracking error, as measured by the standard deviation of the excess returns.
We show how the investor can adjust the performance of naively estimated, tracking-error-optimizing portfolios so that actual portfolio performance is more accurately anticipated. We use global country equity indexes to illustrate the adjustments, construct frontiers of expectation versus standard deviation of portfolio excess return, and test for significant reductions in the out-of-sample bias in the mean. We show that hypothetical transaction costs are lower after we eliminate unnecessary trades driven by estimation noise (see Keim [1999] for an estimate of the trading costs impact for small-cap funds).
THEORETICAL BIAS ADJUSTMENTS
Consider tt > 3 assets with rates of return observed over T+ 1 time intervals, where R denotes the observed (/ rate of return on asset i at time i, and let wB denote the column vector of weights of the n stocks as held in the given benchmark. Asset return vectors R( = (R ,..., RJ' are assumed to be independent and identically normally distributed with unknown true mean vector |a = E (R() estimated unbiasedly at time Tas ji = £^,R(/T, and with unknown true covariance matrix V= Cov(R) estimated unbiasedly as V = yryE^R,- |i)(R( - |i)'. We assume that the elements of (J. are not all equal and that V is nonsingular. Denote the estimation errors as 8 = |i-|i and E = V-V. It is well known that the normality of returns is an unrealistic assumption. However, normality is the preferred approach to allow for mathematical tractability in portfolio computations (see Qian et al. [2007] for a review of issues and why normality is nevertheless the preferred assumption). We address this issue empirically by testing the robustness of the derived formulae via bootstrap simulations.
Let wp denote the vector of weights chosen by the fund, and wg the weights of the benchmark, and let w = wF - wB represent the active weights. Then the fund's realized excess return X at time T + 1 may be written as
... (1)
while the expected value and variance of excess returns (also known as the tracking error), denoted a and C2 respectively, are
... (2)
... (3)
Note that the nature of the excess return, its expectation, and its variance all have the same functional form, regardless of the particular choice of benchmark weights wg.
We are interested in understanding how the estimation errors, 8 and E, affect a mean-variance optimized portfolio. Let ao denote the given target for the expected excess return. Without loss of generality, we assume that the investor has a risk parameter of 1. The classical tracking-error-optimization problem therefore can be written as
... (4)
subject to
... (5)
This optimization problem does not address more realistic constraints on the weights, such as positivity constraints (no short sales) or limits on the number of assets. The optimization problem with more realistic constraints most often does not have a closed-form solution. However, most numerical approaches (such as the sequential programming method, as a popular example) involve solving the optimization problem directionally over a sequence of iterations. Therefore, improvements on the classical optimization problem incorporating estimation error can also have applications to numerical approaches. The results presented here might reasonably be viewed as a lower bound on the effect of estimation error in the case of constrained mean-variance (because these are typically pure mean-variance solutions after the constraints are discovered and imposed, a process that is itself susceptible to the additional effects of estimation error).
The active weights that solve the classical optimization problem may be written as
... (6)
where we define the n X 1 vector 1 = (1, ..., 1)' and the 2x2 matrix B = [(1 Jl)' V 1 (1 Jl)] '. The resulting tracking error is then
... (7)
Substituting the estimates jl and V based on observations at times t = 1, ..., T in place of their true unobservable values Jl and V, we estimate the perturbation weights for a given target expected excess return a as
... (8)
where we use the natural definition .... The weights w would indeed be optimal if jl and V were the true parameters, but are instead suboptimal, as compared with the best possible but unobservable perturbation weights (6).
The next-period excess return (at time T + 1) of a portfolio with estimated perturbation weights w is X = w Rt+v a dot product of two independent random vectors. To characterize this next-period return, we seek adjustments to two naïve excess-return performance measures. We define the naïve expected excess return as the target excess return (X0=û''jl, while the naïve standard deviation of the excess return is
... (9)
Note that ao would be the expected excess return and G0 would be the standard deviation of the excess return if JR7+) were chosen from a distribution with mean jl and covariance matrix V instead of the true parameters Jl and V We also let ... denote the standard deviation of the portfolio, with weights abased on the unobservable true parameters.
Our main results suggest an approximate correction for the bias of these naïve performance measures a0 and G0. By correcting the expected excess return and tracking error for estimation error, the investor can use the adjusted active weights waijusKd to achieve an expected excess return of (X *
... (10)
Adjusting a0 and <T0 for estimation error is not easy, because the expectations involved do not seem to have closed-form mathematical solutions. Following Siegel and Woodgate [2007], we asymptotically approximate jl and V, using the method of statistical differentials (Kotz et al. [1988]).
The main results, presented in theorems 1 and 2, show how an investor can adjust for the bias inherent in the use of estimated values (jl.K) in place of the true asset parameters (JI.K), where these adjustments approximately reflect the future uncertainty of the not-yet-observed time T+ 1 asset returns Rr+) together with the estimation uncertainty in the perturbation weights w.
Theorem 1: The target expected excess return 0C() is systematically biased as a measure of expected excess return, because the delta-method expectation of the excess return is Eà (u/ Jl), where
... (11)
If we define the adjusted expected excess return to be the estimated right-hand side of (11) so that
... (12)
then à,Md is a second-order unbiased estimator of expected excess return.
Theorem 2: The naïve tracking error O,', is systematically biased because its delta-method expectation is
... (13)
while the actual tracking error, evaluated using the delta method, is
... (14)
If we define the adjusted standard deviation to be
... (15)
then G2adjuMll is a second-order unbiased estimator of actual tracking error.
Proofs are in the online appendix.' As expected, the adjusted expected excess return from Equation (12) is less than the target expected excess return 0CÜ when we seek a higher target mean than the benchmark and n > 3, because B is positive definite. Further, the adjusted standard deviation Gadjmtei from Equation (15) always exceeds the naïve standard deviation o0 that would prevail if the true parameters ((X, V) were equal to the estimated parameters (|i, V). Only n and T are used to find the adjusted standard deviation Gdd]muJ from O,,, but in finding the adjusted expected excess return 0.adjuucd from a0, we also use the estimated values JX and V to obtain B22.
These derivations can be used in practice to correct for estimation error ex ante. If the investor uses the formula given in Equation (10), then her adjusted alpha is <XÜ because the nominal alpha a" ...is just enough greater than 0t(| so that, after adjustment, it returns to 0C(|. However, we note that this is accomplished by imposing higher risk on the portfolio. Using the adjustment of Equation (10) lets us also apply the formula derived in numerical iterations used to solve mean-variance optimization problems with non-trivial constraints.
For both the mean and the standard deviation, the bias adjustment is greater when there are more assets n and when there are fewer time periods T. Adjustments are inversely proportional to the number T of observations, which is reasonable because, with more data, we expect the estimates to be more reliable. Adjustments are directly proportional to a linear function of the number n of assets, which is reasonable because, with more assets, there is more flexibility available to mislead while optimizing over the wrong distribution.
EMPIRICAL RESULTS WITH GLOBAL PORTFOLIOS
Because the adjustments are asymptotic, questions arise: How well do the bias adjustments work in finite (small) samples? How significant, statistically and economically, are the adjustments? Our investment opportunity set consists of index portfolios for each of 18 developed countries: Australia, Austria, Belgium, Canada, Denmark, France, Germany, Hong Kong, Italy, Spain, the Netherlands, Norway, Singapore, Spain, Sweden, Switzerland, the U.K., and the U.S. We compute monthly returns from the total return country index with dividends reinvested, obtained from Morgan Stanley Capital International (MSCI), for the 464-month period fromjanuary 30, 1970, to September 30, 2008.
The benchmark used in the studies that follow is an equally weighted portfolio of country returns. The key reason for constructing an equally weighted benchmark rather than using a given capitalization-weighted benchmark is that we want to see how sensitive the adjustment formulae are to small samples, and we don't want a few large countries to dominate the benchmark.
Exhibit 1 presents the descriptive statistics for the excess return data.
We used the Jarque-Bera test to test the normality assumption for each country. The test rejects the hypothesis that excess returns are normally distributed for every country tested, and it indicates that the distributions of excess returns have fat tails, which is a well-known issue.
ESTIMATION ERROR ADJUSTMENT IN GLOBAL EQUITY PORTFOLIOS
Using the full dataset of 18 country indexes, we form portfolios of excess returns using 6 and 18 country indexes, relative to the equally weighted benchmark in each case. Exhibit 2 presents the efficient frontiers of excess returns versus the standard deviation of these excess returns. For the naïve frontier, the 0C(1 naïve excess return is graphed versus the naïve standard deviation given by Equation (9), which ignores the existence of estimation error. Note from Equation (9) that Gn = ^/ß22OC0, so the resulting graph is a straight line with the slope ~JB22 . The adjusted frontier uses the adjusted expected excess return, 0.aijmted from Equation (12), graphed versus the adjusted standard deviation from Equation (15).
With six assets, the naïve frontier is markedly wider than the adjusted frontier, while the adjusted frontier is quite narrow. These indicate that estimation error plays a substantial role even with a small number of assets. As the number of countries increases, both frontiers widen. If we hold 0t() constant across the three charts, we note that both the naïve and adjusted standard deviations decrease as the number of assets increases, indicating that the additional assets have been helpful in controlling risk. In all cases the estimation error adjustment seems to have an economically meaningful impact.
BOOTSTRAP SIMULATIONS
We present a bootstrap simulation study to assess the finite-sample effectiveness of the asymptotic bias adjustment in a situation where we know the true generating distribution for the asset returns. By using bootstrap methodology, our results are robust to nonnormality and avoid the choice of a particular parametric distribution. This is of particular relevance given the Jarque-Bera tests presented in Exhibit 1, showing that excess returns used in this study are non-normal.
We begin with the full data set of 464 months for 18 countries that define the population of months. From this population, we get the bootstrap method samples (with replacement) vectors of country returns. We consider 6 and 18 country portfolios, in each instance drawing all country returns simultaneously from the sampled month, to preserve the empirical covariance structure.
We simulate portfolios using three time intervals: T = 60, 120, and 464 months, choosing in each case 10,000 bootstrap samples. Each sample is a time series using T months, of country index excess returns over the equally weighted benchmark for that number of countries. We average each frontier (actual, naïve, and adjusted) across the 10,000 simulations. We compute the actual frontier using the empirical means and covariances (the exact bootstrap distribution) with the estimated weights. We compute the naïve frontier using the estimated means and covariances with the estimated weights, and compute the adjusted frontier using Equations (12) and (15).
Exhibit 3 presents the bootstrap averages using 60, 120, and 464 months.
With this window of 60 months, there is considerable estimation error with small numbers n of assets (six countries) and the adjustment seems to nearly perfectly correct the bias by aligning the adjusted and actual frontiers. As the number of countries grows to 18, the estimation error remains large and the bias adjustment moves closer to the actual frontier; however, the bias adjustment starts deviating from the actual frontier, as the next asymptotic term in the bias adjustment for the standard deviation is quadratic in n for a given T.
As the number of months grows, the asymptotic bias adjustment improves, with the adjusted frontier becoming indistinguishable from the actual frontier, indicating that the proposed adjustment is robust to the return normality assumption.
ROLLING WINDOW TESTS
We test the statistical significance of the estimation-risk bias adjustment by forming portfolios, each using estimates from a fixed-length data window, then calculating the time series of one-step-ahead performance u',Rf+l, and comparing this time series of actual performance to the expected performance anticipated by the naïve and the adjusted frontiers. At each step in time, using a window of the most recent 36 and 120 months of data, we calculate the mean-variance frontier (with and without adjustment) and estimate the naïve portfolio weights wt corresponding to a fixedtarget excess return of 2% per annum, converted to a monthly target of 2/12% for computing purposes. Generally, we annualize all reported excess returns by multiplying monthly values by 12. We then observe this estimated portfolio's rate of return w,R,+] for the following month. In this way, we obtain three timeseries datasets (actual performance û',R,+1, the constant naïve target performance, and the time-varying, biasadjusted target). We then compute median differences (naïve minus actual and bias-adjusted minus actual) and p-values from the Wilcoxon signed rank test (we chose nonparametric testing methodology to limit the influence of outliers).
Exhibit 4 reports bias estimates and tests for varying numbers of countries and window sizes, measuring the bias as the median difference (in percentage points) of the annualized monthly ex ante anticipated excess return (naïve = monthly equivalent of 2% annual, or adjusted) minus actual monthly performance.
In every case, the adjustment reduces the size of the bias and decreases its significance as measured by the p-values. The naïve anticipated portfolio performance is statistically significantly different from that of the actual performance at all 20 combinations of horizon and sample size, indicating that the effect of estimation error is significant. The adjusted portfolio has non-significant bias in 8 of the 10 cases, tending to show significant bias with more countries and a smaller window size. However, reflecting the asymptotic nature of the bias adjustment, as the estimation window is increased, the bias in the adjusted portfolio becomes non-significant across all country portfolios.
REBALANCING COSTS
One of the issues with forming portfolios while ignoring estimation error is that the portfolios thus formed would theoretically require more frequent rebalancing due to noise in the estimated weights, especially if they are required to be within a given tracking-error band. We analyze the rebalancing costs and size of rebalance for naïve and adjusted portfolios formed by targeting 2% annual excess returns (where the adjusted portfolio targets an adjusted return of 2%). We perform the analysis while varying the number of countries and using estimates from a fixed-length rolling data window.
For both the naïve and adjusted portfolios, we applied the following rebalancing rule. First, the portfolios are rebalanced if their respective standard deviation jumped from one period to the next. We set the jump that triggers a rebalance to the monthly equivalent of a 3.5% annual standard deviation. Second, the portfolios are rebalanced if the realized return is less than the monthly equivalent of one-half the target of 2% (i.e., 1%) annually. Because the monthly volatility can be quite significant, leading to outlier portfolios (monthly standard deviations are in the range of 4% to 7%), we added another rebalancing rule. If the adjusted portfolio has a negative alpha ex ante (as the adjustment would be so large, given the estimation noise, that it changes the sign), then the investor using the adjusted rule will choose to hold the benchmark. As the investor using the naïve portfolio has no ex ante ability to judge whether the 2% annual target is achievable, they cannot implement a similar approach. If the portfolio was not rebalanced, the weights were not changed. If it was rebalanced, the optimal weights for that point in time were computed and used. The investor using the naïve formulae would target 2% annualized, and the adjusted investor would compute the weights that would correspond to an adjusted portfolio return of 2%, using Equation (12).
Exhibit 5 shows the mean, median, standard deviation, information ratio, and /-statistic of the means of the actual portfolio's annualized excess return (for the naïve and adjusted portfolios, respectively). This is to see if the portfolio's excess return relative to the alpha target was higher under the naïve or the adjusted frontier. Then we computed the percent of changes required (i.e., percent rebalances) for the naïve and adjusted portfolios, respectively. This is to see if using the naive rule leads to more adjustments (hence higher costs using a linear transaction-cost estimate) than using the adjustment. We also computed the average size of the rebalancing change required (the sum of the absolute changes in the weights, averaged over all months that involved rebalancing), again to see whether the naive or the adjusted portfolios required larger rebalances (and therefore would be more costly).
The zero median excess returns, for the adjusted rule across all windows, are due to the fact that more than half of the time periods show such a strong adjustment that the adjusted alpha becomes negative. In these cases, the adjusted rule chooses to hold the equally weighted portfolio, leading to excess returns of zero for these periods.
For the shorter window of 36 months, the adjusted portfolios have a higher mean excess return than do the naïve portfolios, with generally higher information ratios, in spite of their generally higher standard deviations. The adjusted portfolio rebalances less frequently, but then adjusts by more (on average) when it does adjust. As the window gets larger, at 60 months the naïve portfolio does better in achieving a higher mean, although that comes at the expense of a negative (worse) median than the adjusted portfolio for 6 and for 18 countries. Standard deviations are higher with naïve than adjusted portfolios for 6 countries and are lower for 12 and 18 countries. The naïve portfolio gets rebalanced more frequently, with a smaller mean adjustment. Neither the naïve nor the adjusted portfolio achieves close to its target, which may be due to a lack of available alpha in the data, which occurs when country means are similar to one another (so that all portfolios have approximately the same mean as the equally weighted benchmark).
We expanded the earlier experiment by increasing the available alpha via scaling the mean returns of the assets to be between -2% and 2% annualized, equally spaced by the number of countries (for example, with 6 countries, country number one would subtract the monthly equivalent of 2% annually from its each of its excess returns, country number two would subtract 1.33% respectively, and so forth). Exhibit 6 presents the results using these newly scaled assets.
Exhibit 6 results show that, with increased alpha added to the data, the adjusted portfolio outperforms the naïve portfolio by having higher mean and higher median across all assets and all windows, although the standard deviations are also higher (which may be necessary in order to come closer to the target alpha of 2%). The rebalances are less frequent and larger (on average) for the adjusted portfolio relative to the naïve portfolio, and the overall impact of transaction costs (taken as percent rebalances times the mean rebalance size, as a rough measure) is generally slightly higher for the adjusted portfolio than for the naïve portfolio.
SUMMARY AND DISCUSSION
We have derived adjustments to reduce and asymptotically eliminate the bias in the performance of tracking-error-optimized efficient portfolios formed in the presence of estimation error. When this approximation is effective, the adjusted efficient frontier more realistically represents the actual performance of a portfolio that seeks a target mean performance, while minimizing the tracking error in a mean-variance framework using the estimated asset distribution. In agreement with results of previous empirical investigations, the size of our theoretical adjustment (for both the mean and the standard deviation) increases linearly with the number of assets (because with more assets, there is more flexibility for statistical error to distort the optimization process) and decreases inversely with the size of the dataset, as measured by the number of time periods (because statistical error tends to decrease in larger samples). Our derivations are based on Taylor-series expansions and are therefore not finite-sample exact. The approximations will tend to degrade when T is smaller (because the statistical estimates will tend to be farther from the true parameter values about which they are expanded) and when n is larger (because the optimization procedure may tend to choose a combination of assets with high estimation error).
Our empirical results illustrate the economic importance of the size of the bias adjustment for international portfolios, the efficacy of the bias adjustment using bootstrap simulations while relaxing the normal approximation, the statistical significance of the bias adjustment, and the slightly higher rebalancing costs incurred while using the adjustment to improve portfolio performance.
ENDNOTE
The authors gratefully acknowledge financial support from the Q-Group and from the University of Washington's department of finance. The opinions expressed in this research paper are not necessarily those held by Russell Investments, its affiliates, or its subsidiaries.
1 See www.iijpm.com.
REFERENCES
Chan, L.,J. Karceski, andj. Lakonishok, "On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model." NBER Working Paper No. 7039, 1999.
Kotz, S., N. Johnson, and C. Read. Encyclopedia of Statistical Sciences (Vol. 8). New York: John R. Wiley & Sons, 1988.
Pachamanova, D. "Handling Parameter Uncertainty in Portfolio Risk Minimization." The Journal of Portfolio Management, Vol. 32, No. 4 (Summer 2006), pp. 70-78.
Scherer, B., and D. Martin. Introduction to Modern Portfolio Optimization with NuOPT, S-PLUS and S+Bayes. New York: Springer, 2007.
Qian, E., R. Hua, and E. Sorensen. Quantitative Equity Portfolio Management. Boca Raton, FL: Chapman & Hall/CRC, 2007.
Siegel, A., and A. Woodgate, "Performance of Portfolios Optimized with Estimation Error," Management Science, Vol. 53, No. 6 (June 2007), pp. 1005-1015.
_____. Online Appendix to "Performance of Portfolios Optimized with Estimation Error." Management Science, Vol. 53, No. 6 (June 2007). http://faculty.washington.edu/asiegel/ OnlineAppendixSiegelWoodgate.pdf.
To order reprints of this article, please contact Dewey Palmieri at [email protected] or 212-224-3675.
ARTEMIZA WOODGATE
is a senior research analyst at Russell Investments in Seattle, WA.
ANDREW F. SIEGEL
is the Grant I. Butterbaugh Professor at the Foster School of Business, University of Washington in Seattle, WA.
APPENDIX
PROOFS
Proof of Theorem 1
The expected excess return of the estimated portfolio, using iterated conditional expectations and the fact that R is independent of the previous observations, may be written as ... Using the method of statistical differentials, the delta-method approximation to this expectation is .... Substituting for w, we find
... (A-1)
Next, we note that ... because 8 does not appear in the covariance estimate, ... by definition of 8, and
... (A-2)
from Lemma 4 of Siegel and Woodgate [Online Appendix, 2007]. This leads to
... (A-3)
Next, we transpose an embedded bracketed scalar to find
... (A-4)
Now, use the fact that E(o'Q~) = tr(QV)/T for any symmetric n x n matrix Q (e.g., see Seber [1984], p. 14) to establish
... (A-5)
Evaluating while using commutativity of matrices within the trace operator, we find
... (A-6)
which completes the proof of Equation (11). To establish Equation (12), we expand Eâ (á^,^) to second order as follows:
... (A-7)
where the last equality was obtained by observing that expectations of second-order expansion terms with respect to either 8 or E will be 0(1/T) by Lemma 1 of Siegel and Woodgate [Online Appendix, 2007],
Proof of Theorem 2
Using the method of statistical differentials, the delta-method approximation to the expectation of the naive tracking error is .... We substitute using the definitions of w and of to find
... (A-8)
We use Equations EC25 and EC26 from Lemma 7 of Siegel and Woodgate [Online Appendix, 2007] together with the fact that O2 = OC2 B22 to obtain
... (A-9)
which establishes Equation (13).
The actual tracking error may be approximated as follows:
... (A-10)
In Equation (A-10), we recognize the first of the four final terms, EA(w'Vw), which has already been evaluated in Equation (13), and use the fact that E(8e) = 0 while expanding the second term of Equation (A-10) as follows:
... (A-11)
Substituting for w and for w, we expand to find
... (A-12)
We use Equations EC8 and EC 13 from Lemmas 3 and 4 of Siegel and Woodgate [Online Appendix, 2007] to obtain
... (A-13)
The expectation in Equation (A-13) may be evaluated using Lemma 1 of Siegel and Woodgate [Online Appendix, 2007], which follows from theorem 3.1 (iii) of Haff [1979], and also using Equation EC22 [Online Appendix, 2007] to evaluate a trace and obtain
... (A-14)
Using this expectation from Equation (A-14) in the formula given in Equation (A-13) for Ea(w'Ew) , the evaluation of the second term of Equation (A-10) is completed as follows:
... (A-15)
We evaluate the third term of Equation (A-10) as follows:
... (A-16)
where the final term of Equation (A-16) was evaluated using Lemma 1 of Siegel and Woodgate [Online Appendix, 2007] along with commutativity of matrix multiplication within the trace function. To complete the evaluation of Equation (A-16), we will also need
... (A-17)
Using EC11 of Siegel and Woodgate [Online Appendix, 2007] we find
... (A-18)
and therefore,
... (A-19)
where we have transposed an embedded scalar in the second term. Applying results from Siegel and Woodgate [Online Appendix, 2007]; Lemma 1 to evaluate the expectations and EC22 for a trace), and using commutativity of matrices within the trace function, we find
... (A-20)
Finally, using this in Equation (A-16) and substituting CTq = (t20B22 we find
... (A-21)
which completes the evaluation of the third term of Equation (A-10). Putting Equations (A-9, A-15, A-16) and (11) together, we can evaluate Equation (A-10) as follows:
... (A-22)
which, together with Equation (13), establishes Equation (14).
To establish unbiasedness, we expand EA(à2,lrJ) to second order as follows:
... (A-23)
where the last equality was obtained by observing that expectations of second-order expansion terms with respect to either 6 or e will be 0(1 IT) by Lemma 1 of Siegel and Woodgate [Online Appendix, 2007], Finally, use Equation (14) to see that
... (A-24)
Thus completing the proof.
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