## Abstract

The author develops the alternative methodology in (1) cross-sectional properties of presumed (economic) factors/proxies that are considered essential for asset returns asymptotically, and (2) test statistics that can be applied to test these cross-sectional properties for empirical asset pricing models. Many model specification tests for these models have emphasized the statistical inferences on time-series properties of estimators and test statistics.

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## Notes

- 1.
Notice that we assume the static factor structure here for simplicity. Further extension of the inter-temporal dependence among the factors as in dynamic factor models can be provided if needed. In addition, with this setting, the orderings of the proxies for factors are not necessarily known in advance (see Ouysse 2006). In other words, the conclusion in the following theorems and claims hold true whether the orderings of proxies for factors are known or unknown.

- 2.
See Cochrane (2001) p. 129 for details.

- 3.
Assuming that \(E[\beta _{i}^{h}]=0\) for all for all

*i*= 1, 2, …*n*is for simplicity. The following arguments in Theorem 4.1 will still hold if a more general condition such as \(E[\beta _{i}^{h}]\neq 0\) for almost all*i*= 1, 2, …*n*is introduced. - 4.
For simplicity, it is assumed that the presumed proxies for factors

*f*_{ t }and hidden factor \(\left \{ f_{ht}\right \} _{i=1,\ldots ,T}\) are orthogonal to each other. - 5.
Specifically, the theoretical asset pricing models with factor structures will need to verify that these factors are compensated with (statistically) significant associated risk premiums. Since no assumption is given on the a priori existence of a factor structure, as in most theoretical asset pricing models, no further discussion on the second-pass regressions are considered here. The intent of model searching is then the identification of any expansion of proxies for factors within the current context of models. In this perspective, the following analyses are aimed at devising model selection with testing through diagnostic tests.

- 6.
The notation here implies that all elements in the vector

*β*_{ H }are square-integrable in*L*^{2}space. - 7.
- 8.
The selection rule as \(\ddot {\xi }(\epsilon _{it})>0\) is chosen as a simplification. Additional rules can be introduced if more detailed descriptions for sample selection are provided.

- 9.
This means that only the population mean of the random factor loadings is equal to zero. It doesn’t state that the hidden factor loadings are equal to zero for all assets of interest.

- 10.
Notice that the notation is only to state that not all factor loadings of the hidden factor are equal to zero.

- 11.
The following section is a modified version of work in Jeng and Liu (2012) which provides details of applications with backward elimination in the model search.

- 12.
In essence, the search is similar to the parsimonious encompassing test.

- 13.
However, a criterion is set that the sampling will only consider the existing firms in the sample periods.

- 14.
One possible reason for using conditional moments is to allow the introduction of a dynamic factor structure when past information is taken into account.

- 15.
Jeng and Liu (2012) provide a cross-sectional long-memory test to verify the possible non-diversifiable hidden factor in the idiosyncratic risk when the Fama-French three-factor model is applied for security returns. The empirical result indicates that the claim of additional hidden factor(s) for the Fama-French model is not confirmed since there is no significant cross-sectional dependence shown in the idiosyncratic risk.

- 16.
Certainly, it will more general to include a random noise

*u*_{ e }in Eq. (4.1.55), where*u*_{ e }is independent of \( \underline {\upsilon }_{i}, X,\) and*f*_{ h }. However, it is easy to see that if*X*_{ e }=*Xγ*^{⋆}+*a*_{ h }*f*_{ h }+*u*_{ e }, the noise can be combined in the hidden factor such that \(a_{h}f_{h}+u_{e}=a_{h}(f_{h}+\frac {1}{a_{h}}u_{e})=a_{h}f_{h}^{\star }\). then, rewrite Eq. (4) as \( \underline {\epsilon _{i}}=\beta _{i}^{h}f_{h}+ \underline {\upsilon }_{i}=\beta _{i}^{h}f_{h}^{\star }+( \underline {\upsilon }_{i}-\frac {1}{a_{h}}u_{e})=\beta _{i}^{h}f_{h}^{\star }+ \underline {\upsilon }_{i}^{\star }.\) The arguments here will still apply. - 17.
The following results still hold if the generalized inverse matrices are applied to the matrices (

*X*^{′}*X*) and \(X_{e}^{\prime }MX_{e},\) respectively. - 18.
In Pesaran (2006), the noises \( \underline {\upsilon }_{i}\) and \( \underline {\zeta }_{i}\) are mutually independent, yet each has serial dependence. For simplicity, we assume they are not of serial dependence here.

- 19.
A possible scenario may also occur when applying Theorem 4.1 that the cross-sectionally weighted average of these long-memory random variables may still contain some long dependence due to the non-diversifiable hidden factor.

- 20.
In fact, it is easy to discover that all these test statistics are all based on the sum of CUSUMs of the underlying variables.

- 21.
The other advantage for the setting is that the test statistics developed here can be applied at times when one is concerned with some possible misspecification errors in the factor structure, as long as the sample size of the cross-sectional observations is sufficiently large.

- 22.
In brief, this cross-sectional detection approach can be applied to out-of-sample statistics for some given time horizons

*T*,*T*→*∞*. Furthermore, if additional decision rules are added, a monitoring scheme can be developed if one is interested in the on-line checking for the hidden non-diversifiable factor. - 23.
Notice that the setting here provides the conditions even for a sequential detection test with additional observations where

*t*=*T*+ 1,*T*+ 2, …, and*T*is the current time horizon used for training samples to estimate the coefficients in the hypothesized models when*T*is sufficiently large. - 24.
Notice that if \(H=\frac {1}{2}\), then

*Z*_{ n }(*j*) will converge in distribution to the quadratic variation of the Brownian motion if Assumption A9 holds. - 25.
Notice that the central limit theorem for these random functions can be extended to the dependence conditions. For simplicity, an independence condition is assumed here.

- 26.

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Jeng, JL. (2018). Finding Essential Variables in Empirical Asset Pricing Models. In: Empirical Asset Pricing Models. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-74192-5_4

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