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英文文献：Unbalanced Regressions and the Predictive Equation-不平衡回归和预测方程
英文文献作者：Daniela Osterrieder,Daniel Ventosa-Santaulària,J. Eduardo Vera-Valdés
英文文献摘要：
Predictive return regressions with persistent regressors are typically plagued by (asymptotically) biased/inconsistent estimates of the slope, non-standard or potentially even spurious statistical inference, and regression unbalancedness. We alleviate the problem of unbalancedness in the theoretical predictive equation by suggesting a data generating process, where returns are generated as linear functions of a lagged latent I(0) risk process. The observed predictor is a function of this latent I(0) process, but it is corrupted by a fractionally integrated noise. Such a process may arise due to aggregation or unexpected level shifts. In this setup, the practitioner estimates a misspecified, unbalanced, and endogenous predictive regression. We show that the OLS estimate of this regression is inconsistent, but standard inference is possible. To obtain a consistent slope estimate, we then suggest an instrumental variable approach and discuss issues of validity and relevance. Applying the procedure to the prediction of daily returns on the S&P 500, our empirical analysis confirms return predictability and a positive risk-return trade-off.
具有持续回归的预测回归通常受到(渐近)偏差/不一致的斜率估计、非标准的甚至可能是虚假的统计推断和回归不平衡的困扰。我们通过建议一个数据生成过程来缓解理论预测方程的不均衡性问题，其中收益作为滞后潜在I(0)风险过程的线性函数生成。观测到的预报器是这个潜在的I(0)过程的函数，但它被一个小积分噪声所破坏。这样的过程可能是由于聚合或意外级别转移而出现的。在此设置中，专业人员估计了一个不指定的、不平衡的、内生的预测回归。我们表明，这个回归的OLS估计是不一致的，但标准推断是可能的。为了得到一个一致的斜率估计，然后我们建议一个工具变量的方法，并讨论有效性和相关性的问题。将该程序应用于标准普尔500指数日收益的预测，我们的实证分析证实了收益的可预测性和正的风险收益平衡。
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