By means of Monte Carlo simulations for returns generated from an artificial capital market, we illustrate below how and to what extent different proxies of the market index influence the quality of the CAPM parameter estimates. By simulating from a well-defined artificial capital market, for which the CAPM holds, we can define market proxies as the outcome of the true data generating process combined with misspecified weights rather than imposing arbitrary stochastic assumptions on the error process.

Our simulation study is based on 10,000 Monte Carlo samples of monthly excess return series for an asset universe of *N* = 205 assets over 10 years. The data generating process for the excess returns is given by:
$\begin{array}{r}{r}_{t+1}=\gamma \mathrm{\Omega}{b}^{\ast}+{\epsilon}_{t+1},\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{\epsilon}_{t+1}\stackrel{\mathrm{i}\mathrm{i}\mathrm{d}}{\sim}{N}\left(0,\mathrm{\Omega}\right).\end{array}$

The variance-covariance matrix *Ω* was chosen to be equal to the sample variance-covariance matrix calculated from monthly data on excess returns of 205 components of S&P500 index from January 1, 1974 to May 1, 2015. For these 205 stocks a sufficiently long time series was available to obtain a reliable estimate of the high-dimensional covariance matrix *Ω*. In order to use realistic values for the true weight vector ${b}^{\ast}$ we use their empirical counterparts. More precisely, for each of the 205 stocks of the S&P500 index we compute the mean value of the market capitalization based on monthly data from January 1, 1974 till May 1, 2015 and define the true market weights as a proportion of the total market capitalization of the 205 stocks. Finally, the coefficient of risk-aversion *γ* was chosen to be equal to 0.4. Following our baseline model, we assume a fixed supply of assets as assumed in eq. (2) with the true market index, ${r}_{m,t+1}^{\ast}$ generated according to eq. (4). In a second step, we generate proxies of the true market index under different types of measurement error. summarizes the five different weighting strategies for the market proxy used in the Monte Carlo study.

Table 1: Functional form of misspecified weights *b* of market indices.

For the case of random measurement errors, we consider three different choices for the variance of the measurement error, ${\sigma}_{\nu}=0.05,\phantom{\rule{thinmathspace}{0ex}}0.025,\text{and}0.01$. For our analysis of the market proxies based on the subsets of the asset space, we choose subsets covering 18 %, 25 % and 50 % of the total market capitalization. For these subsets, only assets with the largest weights are selected, so that the indices are more comparable to real-world market indices. We estimate the CAPM parameters for 15 randomly drawn assets by the seemingly unrelated regression (SUR) method. summarizes our findings for market proxies based on the total asset space and random error in the weighting scheme, while contains the results for indices based on specific subsets. Both tables summarize our findings by reporting the means of the estimates for 15 selected assets. The detailed results for each of the 15 assets are given in in Appendix A.3.

Table 2: MC-results for the CAPM with random measurement error (CAPM-RME).

Table 3: MC-results for the CAPM with fixed measurement error.

For reasons of comparison, the second column of contains the results of the CAPM, when the true market return is feasible. Since these estimates are obtained under the true data generating process, they only differ from the true model parameters by the sampling error. Therefore, these estimates can serve as a benchmark for the estimates using market proxies. Under the true data generating process, the CAPM alphas are on average close to their theoretical value of zero. The empirical rejection rate for ${\stackrel{\u02c6}{\alpha}}_{j}$ is close to the 5 % significance level, which indicates that the sample size of *T* = 120 chosen for the Monte-Carlo simulations is sufficiently large to produce estimates that come close to the true parameters given the distributional assumptions and the true market index. We also report the results for the joint test of absence of abnormal returns. Since the errors in the simulations are normally distributed, the F-test is the appropriate choice for a finite sample test. We also report the Wald statistics based on the true *Ω* to avoid finite sample distortions going along with Wald test based on large dimensional estimated covariance matrices. For the true CAPM model without measurement error both tests show an empirical rejection rate close to the nominal 5 % significance level, so that we can conclude that two joint tests do not suffer from any finite sample deficiencies that may distort the test results for the models based on proxies of the market index.

It is important to note, that in the case of the CAPM with measurement error the alternative hypothesis of non-zero intercepts is true, i.e. asymptotically the Type I error increases to unity and the power approaches unity as the Type II error vanishes. Therefore, the stronger the measurement error the more the model’s intercepts deviate from the null hypothesis, which explains why in the empirical rejection rates increase with increasing ${\sigma}_{\nu}$.

Only for the case of a small measurement error, ${\sigma}_{\nu}=0.01$, the CAPM-RME shows negligible distortions of the parameter estimates. The empirical correlation between the true market returns and the proxy is 0.96 and the attenuation bias reflected by $\stackrel{\u203e}{\stackrel{\u02c6}{{\lambda}_{1}}}$ is small. For this mild case of measurement error, we find an empirical rejection rate of the null of no abnormal returns of 8 %. For the intermediate case with ${\sigma}_{\nu}=0.025$ the correlation between true market return and the market proxy appears to be rather high with $\stackrel{\u02c6}{\rho}({r}_{m,t}^{\ast},{r}_{m,t})=0.81$. However, for more than 47 % of our estimates we find abnormal returns mimicking the existence of potential profits from trading of the average size of 2.7 % per month. For the case of a large measurement error, the situation deteriorates even more, although the correlation between true market return and the proxy remains above 0.5.

All three scenarios demonstrate that the correlation between the true market return and the market proxy provides insufficient information to make any conclusion about the bias in the CAPM estimates. Contrary to Prono (2015), our results indicate that, after all, what matters is the coefficient on the linear projection ${\lambda}_{1}$, which consists of the product of the square root of the reliability ratio and the correlation coefficient.^{5} The situation does not improve when multi-factor models instead of the CAPM are considered. In multiple regression models with measurement error in one variable the distortions spill over to the estimates of the other parameters and produce size distortions (see Brunner/Austin 2009).

Moreover, the measurement error also has a considerable impact on the precision of the parameter estimates. Compared to the RMSE for the benchmark model, the RMSE for beta increases by 40 % in the case of a small measurement error and almost doubles for the medium size measurement error. The loss in estimation precision has serious implications for concrete applications and interpretations of the CAPM estimates. For instance, the increase of the RMSE due to the measurement error also increases the risk of a faulty sorting into defensive and aggressive stocks. For example, in the case of the medium size measurement error based on the empirical rejection rate of the null hypothesis ${\beta}_{j}<1$, we detect that only 2 stocks have a beta coefficient significantly larger than 1, while in the case of no measurement error we detect 5 aggressive stocks. A larger variance-covariance matrix of the parameter estimates also reduces the power of detecting cumulative abnormal returns based on the CART-test (see Campbell et al. (1997), Chapter 4).

contains the results for the case of fixed measurement errors, where the market proxies are based on a subset of the asset universe. We consider seven different scenarios with fixed measurement errors. The first three cases (columns 2–4) capture the case, where the market proxy is based on the normalized true weights for the largest assets that cover the 18 %, 25 % and the 50 % of total market capitalization, respectively. All indices are constructed such that they include only the largest fraction of assets, while ignoring the corresponding smaller assets of the asset universe.^{6}

In terms of the number of constituents, the indices are based on the 4, 7 and 20 largest stocks of asset universe. This rather small number of included stocks may seem unreasonable at the first glance. However, one should keep in mind that the size of the true market universe in our simulation contains only 205 stocks. In empirical studies, the proxies most often used are the Dow-Jones Industrial Average Index, the S&P500 index and CRSP index. Correspondingly, these indices cover 30, 500 and nearly 4,000 stocks traded on the U.S. stock market. In terms of market capitalization, the S&P500 covers roughly 76 % of the total market capitalization of the 5,200 actively traded companies traded on the NYSE and the NASDAQ. This, however, is far from the capitalization of the entire true market portfolio.

The last four columns of contain the estimates for models, where the true weighting scheme is replaced by equal weights. In the last column the estimation results are given, when the index is based on all assets but equal weights of size 1/205 are used instead of ${b}^{\ast}$. These estimates come rather close to the ones obtained when the true market index is used. Both the reliability ratio as well as the correlation between the true index and the proxy are close to one. This particular finding, however, should not be overemphasized, because it is strongly based on the underlying Monte-Carlo design. The true weights and the equal weights of size 1/205 do not differ very much, so that on average the fixed measurement error turns out to be rather small. The results show the monotonicity of the quality of the estimates in terms of the number of assets used. The estimates with the index covering 50 % of total market capitalization are slightly superior to the estimates based on the other two scenarios. If the index represents only the capitalization of 18 % of the market, the estimates reveal the largest biases and mimic on average abnormal returns of 3.9 %.

Our simulations based on the equally weighted index generate a specific correlation pattern between the estimated alphas and betas depending on the number of stocks included in the index. This pattern is depicted in Figure 1. When more than 60 of largest stocks out of the 205 stocks are included, we find a negative correlation between the estimated alphas and betas with a maximum negative correlation for the case when all assets are included with equal weights. What appears to be evidence for the low risk puzzle is in our case simply the outcome of the specific measurement error of the market index. In fact, Frazzini and Pedersen (2014) use in their study on low risk premium the same construction of the market index.

Figure 1: Correlation between estimated alphas and betas as a function of the number of assets included in the equally weighted market index.

In Figure 2 the top left and top right plots give the empirical rejection rates of no abnormal returns for indices based on different subsets of the asset space. For the case based on fixed measurement error with normalized weights, the empirical rejection rates decrease the more representative the market proxy becomes for the entire market, i.e. the over-rejection rate for the CAPM alphas and the evidence in favor of (spurious) abnormal returns decreases with an increasing fraction of total market capitalization captured by the market proxy. For scenarios, where the market proxy captures 10 % or less of the total market capitalization, the empirical rejection rate strongly exceeds the nominal 5 % level. Note, that for the equally weighted index we do not observe such a strict monotonicity (see upper right panel). Here we find an optimal number of stocks that minimizes the difference between the empirical and the nominal rejection rate. For our design, this is obtained, if around 30 % of the largest stocks are constituents of the index. Equally weighted indices including more stocks turn out to be less representative and proxy the true market index less well.

Figure 2: Empirical rejection rate of the absence of abnormal returns in the CAPM with the index based on the subset of the largest assets.

The Receiver Operating Characteristics (ROC) curves depicted in the bottom right and bottom left plots of Figure 2 provide more insight into the relation between Type I error and power of the tests for absence of abnormal returns depending on crudeness of the market proxy. For the least representative index with *M* = 4 the probability of falsely detecting abnormal returns due to measurement error is substantially larger than for the other two cases.^{7}

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