# 金融代写|利率理论代写Portfolio Theory代考|FIN586 ESTIMATING MODELS WITH EXCESS RETURNS

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## 金融代写|利率理论代写Portfolio Theory代考|ESTIMATING MODELS WITH EXCESS RETURNS

When excess returns $\left(R^{c}\right)$ are used to estimate and test asset pricing models, the moment conditions (pricing equations) are
$$E\left(m R^{e}\right)=0_{N} .$$
Let $m=\theta_{0}-\left(\theta_{1} f_{1}+\cdots+\theta_{K} f_{K}\right)$. In this case, the mean of the SDF cannot be identified or, equivalently, the parameters $\theta_{0}$ and $\left(\theta_{1}, \ldots, \theta_{K}\right)$ cannot be identified separately. This requires a particular choice of normalization. One popular normalization is to set $\theta_{0}=1$, in which case $m=1-\left(\theta_{1} f_{1}+\cdots+\theta_{K} f_{K}\right)$. An alternative (preferred) normalization is to set $\theta_{0}=1+\theta_{1} E\left(f_{1}\right)+\cdots+\theta_{K} E\left(f_{K}\right)$, in which case $m=1-\theta_{1}\left[f_{1}-E\left(f_{1}\right)\right]-\cdots-\theta_{K}\left[f_{K}-E\left(f_{K}\right)\right]$ with $E(m)=1$. These two normalizations can give rise to very different results (see Kan and Robotti, 2008; Burnside, 2010)

Kan and Robotti (2008) argue that when the model is misspecified, the first (raw) and the second (de-meaned) normalizations of the SDF produce different GMM estimates that minimize the quadratic form of the pricing errors. Hence, the pricing errors and the $p$-values of the specification tests are not identical under these two normalizations. Moreover, the second (de-meaned) specification imposes the constraint $E(m)=1$ and, as a result, the pricing errors and the HJ-distances are invariant to affine transformations of the factors. This is important because in the first normalization, the outcome of the model specification test can be easily manipulated by simple scaling of factors and changing the mean of the SDF. This problem is not only a characteristic of linear SDFs but also arises in nonlinear models. The analysis in Burnside (2010) further confirms these findings and links the properties of the different normalizations to possible model misspecification and identification problems discussed in the previous two subsections.

## 金融代写|利率理论代写Portfolio Theory代考|CONDITIONAL MODELS WITH HIGHLY PERSISTENT PREDICTORS

The usefulness of the conditional asset pricing models crucially depends on the existence of some predictive ability of the conditioning variables for future stock returns. While a large number of studies report statistically significant coefficients for various financial and macro variables in in-sample linear predictive regressions of stock returns, several papers raise the concern that some of these regressions may be spurious. For example, Ferson, Sarkissian, and Simin (2003) call into question the predictive power of some widely used predictors, such as the term spread, the book-to-market ratio, and the dividend yield. Spurious results arise when the predictors are strongly persistent (near unit root processes) and their innovations are highly correlated with the predictive regression errors. In this case, the estimated slope coefficients in the predictive regression are biased and have a nonstandard (nonnormal) asymptotic distribution (Elliott and Stock, 1994; Cavanagh, Elliott, and Stock, 1995; Stambaugh, 1999). As a result, $t$-tests for statistical significance of individual predictors based on standard normal critical values could reject the null hypothesis of no predictability too frequently and falsely signal that these predictors have predictive power for future stock returns. Campbell and Yogo (2006) and Torous, Valkanov, and Yan (2004) develop valid testing procedures when the predictors are highly persistent and revisit the evidence on the predictability of stock returns.

Spuriously significant results and nonstandard sampling distributions also tend to arise in long-horizon predictive regressions, where the regressors and/or the returns are accumulated over $r$ time periods so that two or more consecutive observations are overlapping. The time overlap increases the persistence of the variables and renders the sampling distribution theory of the slope coefficients, $t$-tests and $R^{2}$ coefficients, nonstandard. Campbell (2001) and Valkanov (2003) point out several problems that emerge in long-horizon regressions with highly persistent regressors. First, the $R^{2}$ coefficients and $t$-statistics tend to increase with the horizon, even under the null of no predictability, and the $R^{2}$ is an unreliable measure of goodness of fit in this situation. Furthermore, the $t$-statistics do not converge asymptotically to well-defined distributions and need to be rescaled to ensure valid inference. Finally, the estimates of the slope coefficients are biased and, in some cases, not consistently estimable. All these statistical problems provide a warning to applied researchers and indicate that the selection of conditioning variables for predicting stock returns should be performed with extreme caution.

# 利率理论代写

## 金融代写|利率理论代写PORTFOLIO THEORY代 考|ESTIMATING MODELS WITH EXCESS RETURNS

$$E\left(m R^{e}\right)=0_{N}$$

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