This paper examines the use of proxies (or reference variables) for the true factors in the arbitrage pricing theory (APT). It generalizes other authors' existing work and shows that, when there are more reference variables than the true factors, the APT still holds. The possibility of fewer reference variables than the true factors is also considered, but the APT is not shown to hold, in the same sense, for this case. This work builds on an earlier paper by Ingersoll (Ingersoll J 1984 J. Finance 39 1021-39), and our propositions can be thought of as specializations of his theorems. SImilar to Nawalkha (Nawalkha S 1997 J. Financial Economics 46 357-81), our work does not use the mathematics of Hilbert and Banach spaces and, thus, is open to a much wider audience. The practical implication of our results is that model builders should be generous with the number of factors they use, as excessively parsimonious models suffer from inaccuracy.
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机译：本文考察了套利定价理论（APT）中真实因素的代理（或参考变量）使用。它概括了其他作者的现有工作，并表明，当参考变量多于真实因素时，APT仍然成立。还考虑了比真实因素更少的参考变量的可能性，但是在这种情况下，APT在同样的意义上没有得到保持。这项工作建立在Ingersoll的早期论文（Ingersoll J 1984 J. Finance 39 1021-39）上，我们的命题可以被认为是他的定理的专业化。从Nawalkha到SImilar（Nawalkha S 1997 J. Financial Economics 46 357-81），我们的工作没有使用希尔伯特和巴纳赫空间的数学原理，因此对更广泛的受众开放。我们的结果的实际含义是，模型构建者应在使用的因素数量上大方，因为过于简约的模型会出现误差。
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