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Misspecification and Weak Identification in Asset Pricing
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Over the past decades, hundreds of risk factors have been proposed to explain the cross-section of asset returns, making up the, so-called, zoo of factors; see, e.g., Cochrane (2011) and Harvey, Liu, and Zhu (2016).\nocite {harvey2016and} \nocite{cochrane2011presidential} Much of the empirical support for these factors is based on the beta representation, where risk premia are identified by projecting expected returns on the betas, and the betas are the factor loadings in the time-series regression of asset returns on risk factors; see, e.g., Fama and MacBeth (FM, 1973).\nocite{fm73} The resulting FM $t$-statistic on risk premia, \nocite{han82} Hansen (1982)'s $J$ -statistic for misspecification and the cross-sectional $R^{2}$, are popularly used in the asset pricing literature to show support for proposed risk factors. The reliability of these conventional statistics has, however, recently been brought into question by two empirically relevant issues: misspecification and weak identification.
Up till now, misspecification has been widely acknowledged as an inherent feature of asset pricing models; see, e.g., Gospodinov, Kan, and Robotti (2014). \nocite{gkr14} For the beta representation, misspecification results in expected returns that are not fully explained by the betas of the specified factors. Thus, pricing errors generally exist and should be taken into account when conducting inference on risk premia. Kan, Robotti, and Shanken (KRS, 2013) \nocite{krs13} therefore propose the KRS $t$-test, which explicitly allows for the existence of misspecification while the FM $t$ -test does not. As warned by Kan, Robotti, and Shanken (2013), failure to account for misspecification tends to enlarge the $t$-statistic on risk premia, leading to overstated pricing performance of risk factors.
Weak (or no) identification, on the other hand, is driven by poor quality risk factors or more generally, limited information contained in the data. Kan and Zhang (1999)\nocite{kz99}, for example, warn that risk factors proposed in the literature could be useless factors with zero betas. When betas are zero, risk premia are unidentified in the beta representation and paired with misspecification, the FM $t$-statistic can be spuriously large to support useless factors as shown by Kan and Zhang (1999). In addition, Kleibergen (2009) \nocite{kf09} illustrates the malfunction of the FM $t$ -statistic caused by factors that are only weakly correlated with asset returns and proposes robust tests to address weak identification of risk premia. Moreover, Kleibergen and Zhan (2015, 2021) \nocite{kz15} show that the cross-sectional $R^{2}$ and $J$-statistic may also spuriously favor statistically weak factors, respectively.
Despite that misspecification and weak identification are well recognized, there is currently no unified approach for identification and inference that can deal with these empirically relevant issues. We therefore advocate such an approach and apply it to show the widespread occurrence of misspecification and weak identification in applied asset pricing studies.
When misspecification is present, there is no longer a (true) value of the risk premia at which the pricing error is zero. Different estimation procedures then lead to distinct pseudo-true values of risk premia, which are the minimizers of the population objective function of the respective estimation procedure. We show that, in case of generic misspecification, the difference between the pseudo-true value of two estimation procedures, i.e. the FM two-pass approach and the continuous updating estimator (CUE)\ of Hansen, Heaton, and Yaron (1996),\nocite{hhy96} and the baseline risk premia that would apply in case of correct specification, depends on the strength of misspecification compared to the strength of identification. The strength of misspecification is reflected by the population equivalent of the $J$ -statistic, while the strength of identification is revealed by the, so-called, identification strength ($IS$) statistic. The $IS$-statistic is a rank test statistic on the $\beta $ matrix and is, by construction, always larger than or equal to the $J$-statistic. The difference between these two measures is indicative of whether we can interpret the pseudo-true value as a risk premium. When this difference is small, the pseudo-true value basically results from the close to reduced rank value of the $\beta $ matrix, so it can be far off from the baseline risk premium under correct specification, and does not reflect a risk premium. The sample equivalents of the $J$ and $IS$ statistics are therefore important in gauging whether we can interpret risk premia estimates resulting from empirical studies as genuine risk premia of interest.
For eight prominent empirical studies in asset pricing: Fama and French (1993), Jagannathan and Wang (1996), Yogo (2006), Lettau and Ludvigson (2001), Savov (2011), Adrian, Etula, and Muir (2014), Kroencke (2017) and He, Kelly, and Manela (2017) and for all (14 billion+) one to six factor specifications resulting from the factor zoo of Feng, Giglio, and Xiu (2020), we compute the $J$ and $IS$ statistics to illustrate the widespread co-existence of misspecification and weak identification. The resulting $J$ -statistics are mostly large and significant, and if they are not, their smaller values are often induced by an accompanying small value of the $IS$ -statistic. The smallish $J$-statistic then basically results from a close to reduced rank value of the $\beta $ matrix, so the risk premia estimates do so as well and do not reflect risk premia. When we do not incorporate the zero-$\beta $ return or use more time series observations, these identification issues become less problematic for some settings, such as the three-factor model from Fama and French (1993), but remain for many others.
The conventional test statistics for conducting inference on risk premia, such as the FM two-pass $t$-test, also with the Shanken (1992)\nocite{sh92} correction of the standard errors, and the KRS $t$-test, become unreliable in the presence of misspecification and/or weak identification. We illustrate this by showing that the limit behavior of the FM\ two-pass estimator consists of four components. Two of these components are negligible in case of strong factors and correct specification, but lead to non-standard behavior in case of misspecification and/or weak identification. It is also not possible to correct for them by using the bootstrap. We therefore use the double robust Lagrange multiplier (DRLM) statistic from Kleibergen and Zhan (2021) that provenly remains reliable in case of misspecification and/or weak identification. For ease of exposition, we conduct a small simulation experiment to illustrate the pros and cons of these different statistics for testing hypotheses on risk premia.
We illustrate the practical usage of the DRLM test by using the influential Fama and French (1993) three-factor model \nocite{ff93} and the conditional consumption capital asset pricing model in Lettau and Ludvigson (2001). \nocite{letlud01} For the Fama and French (1993) three-factor model, we show that for commonly used data sets, such as those resulting from Lettau and Ludvigson (2001) and Lettau, Ludvigson, and Ma (2019), identification of the risk premia can be problematic when incorporating the zero-$\beta $ return because of the near constancy of the $\beta $'s associated with the market return. This identification issue is alleviated when using more time series observations or by removing the zero-$\beta $ return. The joint 95% confidence set for the risk premia resulting from the DRLM\ test is then bounded (ellipsoid), while it is unbounded in the direction of the risk premia on the market return when using fewer observations and also incorporating the zero-$\beta $ return. For the conditional capital asset pricing model stemming from Lettau and Ludvigson (2001), the 95% confidence sets of the risk premia resulting from the DRLM\ test are all unbounded, indicating limited information in the data for precisely identifying the risk premia. The 95% confidence sets from the DRLM\ test are, for all of these settings, fully in line with the $J$ and $IS$ statistics. When the $J$ and $IS$ statistics are relatively close, we obtain unbounded 95% confidence sets from the DRLM test, in contrast with the bounded ones resulting from the FM two-pass and KRS $t$-tests which are then unreliable. Also the CUE differs considerably from the FM two-pass estimator in this scenario. When the $J$ and $IS$ statistics are substantially apart, the results from all these procedures are, as expected, rather similar, but the 95% (projected) confidence sets from the DRLM\ test are notably narrower than those from the FM two-pass and KRS $t$-tests. It all shows the importance of jointly using the $J$ and $IS$ statistics to gauge the identification of risk premia while using the DRLM\ test to conduct inference on risk premia.
Overall, our study adds to an emerging body of research that aims to bring discipline to the zoo of factors. Harvey, Liu, and Zhu (2016), for example, propose a higher hurdle such as a $t$-statistic greater than 3.0 instead of the commonly used 1.96, since they are concerned that significant $t$ -statistics documented in the existing literature could result from data mining. Unlike Harvey, Liu, and Zhu (2016) and the follow-up work, we do not provide new hurdles of $t$-statistics; instead, we suggest the DRLM, $J$ and $IS$ statistics for gauging the quality of risk factors. Feng, Giglio, and Xiu (2020) \nocite{feng2020taming} focus on whether a proposed risk factor adds explanatory power beyond the existing zoo of factors, so their methodology builds on a high-dimensional set of existing factors. In contrast, our approach is suitable for evaluating the explanatory power of each factor model individually. In order to jointly address misspecification and no identification, Gospodinov, Kan, and Robotti (2014) propose a model selection procedure to eliminate potentially useless factors. Unlike Gospodinov, Kan, and Robotti (2014), our proposed DRLM test aims to infer risk premia regardless of whether misspecification and weak identification are present.
Finally, we note that misspecification and weak identification are not limited to the beta representation; see, e.g., Stock and Wright (2000) and Hansen and Lee (2021), \nocite{sw00} \nocite{hl21} who have studied weak identification and misspecification in the generalized method of moments framework, respectively. Given that asset pricing models are at best approximations of reality while lots of risk factors have little explanatory power for asset returns, misspecification and weak identification are a likely common threat to empirical asset pricing studies.
The rest of the paper is organized as follows. Section (ref) starts with discussing the setup of misspecification and the consequences it has for commonly used risk premia estimators. It illustrates that the commonly used FM $t$-test is jeopardized by both misspecification and weak identification, while the DRLM test takes both issues into account. Section (ref) uses the $J$ and $IS$ statistics to highlight the prevalence of misspecification and weak identification in existing studies. Section (ref) contains the empirical applications for conducting inference on risk premia, while Section (ref) concludes. Technical details are relegated to the Appendix.
Let $R_{i,t}$ be the return on the $i$-th asset at time $t,$ with $ i=1,...,N, $ and $t=1,...,T.$ The beta representation of expected returns models it as linear in the beta vector of factor loadings:
where $\lambda _{F}$ is the $K\times 1$ vector of risk premia, and $\beta _{i}$ is the $K\times 1$ vector of factor loadings:
with $F_{t}$ the $K\times 1$ vector of the specified risk factors, and $ K<N+1 $. We can as well represent the beta representation jointly for all assets by stacking the $N$ equations of ((ref)) to get
with $R_{t}=(R_{1,t},...,R_{N,t})^{\prime },$ $\beta =(\beta _{1},...,\beta _{N})^{\prime }.$
In both ((ref)) and its misspecification extension provided later on (i.e., Equation ((ref))), identification of $\lambda _{F}$ relies on the quality of $\beta _{i}$. For instance, if some of the specified risk factors are just useless noise with zero betas, then $\lambda _{F}$ is unidentified in the beta representation. This reasoning generalizes to the full rank condition of the $N\times K$ -dimensional matrix $\beta ,$ i.e. for $\lambda _{F}$ to be identified, $ \beta $ needs to have full rank. In the single factor case with $K=1$, this rank condition then just requires that $\beta $ should be non-zero. If the full rank condition of $\beta $ is jeopardized by risk factors of poor quality, then $\lambda _{F}$ is potentially weakly identified; see, e.g., Kan and Zhang (1999),\nocite{kz99} Kleibergen (2009),\nocite{kf09} and Kan, Robotti, and Shanken (2013)\nocite{krs13} for a related discussion.
Remark 1: The zero-$\beta ,$ $\lambda _{0}=0,$ restriction. A scalar $\lambda _{0}$ is often added to ((ref) ) so
where $\iota _{N}$ is the $N\times 1$ vector of ones, and $\lambda _{0}$ is the, so-called, zero-beta return, or the expected return to an asset with no exposure to priced risks. Since our interest mainly lies in $\lambda _{F}$, we opt to focus on ((ref)) instead of ((ref)) for ease of exposition. This treatment is related to the $\lambda _{0}=0$ restriction, so that ((ref)) reduces to ((ref)). The zero restriction can be achieved by considering $R_{t}$ as the excess return. Our discussion on ((ref)), however, can be straightforwardly extended to incorporate $\lambda _{0}$. For example, the full rank condition of $ \beta $ for ((ref)) extends to the full rank condition of $(\iota _{N}$ $\vdots $ $\beta )$ for ((ref)) once $\lambda _{0}$ is allowed for.
Remark 2: Incorporating the zero-$\beta $\ return. If $ \lambda _{0}=0$ is not assumed, we can still use ((ref)) to focus on $\lambda _{F}$ by considering $ R_{t}$ as the return in deviation of a reference asset. To illustrate, let $ \mathcal{R}_{t}=(\mathcal{R}_{1,t}\ldots \mathcal{R}_{N+1,t})^{\prime }$ be the $(N+1)\times 1$ vector of returns such that
where $\mathcal{B}$ is the $(N+1)\times K$-dimensional matrix of factor loadings. By subtracting the $(N+1)$-th asset return, we obtain the $N\times 1$ vector $R_{t}:$ $R_{t}=(\mathcal{R}_{1,t}\ldots \mathcal{R} _{N,t})^{\prime }-\iota _{N}\mathcal{R}_{N+1,t}$ such that ((ref)) implies ((ref)), with $ R_{t}=J_{N}\mathcal{R}_{t}$, $\beta =J_{N}\mathcal{B}$, and $J_{N}=(I_{N}$ $ \vdots $ $-\iota _{N})$. Thus, the full rank condition of $(\iota _{N+1}$ $ \vdots $ $\mathcal{B)}$ in ((ref)) is equivalent to the full rank condition of $\beta $ in ((ref)). Our subsequent analysis is invariant with respect to the choice of the $(N+1)$ -th asset; see Kleibergen and Zhan (2020).\nocite{kz20}
Remarks 1 and 2 above show that we can focus on ((ref)) to illustrate inference on $\lambda _{F}$ regardless of whether a zero-$\beta $ return is incorporated or not. \footnote{ It is worth noting that whether $\lambda _{0}$ should be excluded already sheds light on misspecification and weak identification in asset pricing. On the one hand, if $\lambda _{0}$ is non-zero, then incorrectly imposing the zero restriction leads to a misspecified condition. On the other hand, if $ \lambda _{0}$ is zero, then including the redundant intercept term in ((ref)) potentially weakens the identification of risk premia. In particular, when there is little cross-sectional variation in $\beta $, including $\iota _{N}$ causes near-multicollinearity in $(\iota _{N}$ $\vdots $ $\beta )$, which will further induce the weak identification of $\lambda _{F}$. Misspecification and weak identification are, however, more general than whether to impose $\lambda _{0}=0$ or not.}
Under misspecification, the expected returns are not fully explained by the specified $\beta $, so $\mu _{R}\neq \beta \lambda _{F}$. We therefore consider the correctly specified setting above as the baseline, and introduce the pricing error $\tilde{e}$ in ((ref)) to reflect misspecification. In particular, we assume the pricing error to be potentially of a different order of magnitude:
where $a$ is a normalized $N$-dimensional vector, so $O(\tilde{e})$ reflects the magnitude of misspecification; similarly, $b$ is a normalized $N\times K$ -dimensional matrix, so $O(\beta )$ reflects the magnitude of identification.
The specification in ((ref)) is without loss of generality and, for example, allows for misspecification which is much smaller than the expected return resulting from the beta representation, or proportional to it which would result in case of weak identification. To capture this, the magnitudes can be modelled to be proportional to the number of time series observations as is common in the literature on weak identification, for example, $O(\tilde{e})=O(T^{c_{\tilde{e}}}),$ $O(\beta )=O(T^{c_{\beta }})$, $c_{\tilde{e}}$ and $c_{\beta }$ are finite constants; see, e.g., Staiger and Stock (1997)\nocite{stst97} and Kleibergen (2009). \nocite{kf09}
The misspecification is generic, so the difference from the baseline pricing error, $\tilde{e},$ lies both in the space spanned by $\beta $ and outside of it. It captures the notion of the correctly specified setting as a baseline, on top of which there is an unstructured misspecification component. It resembles the estimation errors which the sample moment equations add to the population moment equations. These errors are also unstructured, and lie both in the space spanned by $\beta $ and outside of it. The magnitude components, $O(\tilde{e})$ and $O(\beta ),$ can then further be such that the estimation errors in the sample moment equations are of the same order of magnitude as the misspecification and identification strengths.
Next, we show that the relative magnitudes of misspecification and identification, $O(\tilde{e})$ and $O(\beta ),$ drive the identification of risk premia in the FM two-pass methodology and the CUE. Since the misspecification $J$-statistic gauges $O(\tilde{e})$ while the identification $IS$-statistic reflects $O(\beta ),$ the comparison of these two statistics plays a crucial role in our later discussion.
The population objective functions of different risk premia estimators are all quadratic forms of the pricing error but weigh it differently. The population objective function of the FM two-pass estimator involves no weight function and is therefore just the quadratic form of the pricing error:
having as its minimizer the, so-called, pseudo-true value of risk premia. Unlike the true value in case of correct specification, the pseudo-true value denoted by $\lambda _{F,FM}^{\ast }$ does not set the objective function to zero as $\mu _{R}=\beta \lambda _{F}+\tilde{e},$ with $\tilde{e} \neq 0:$
Using the specification of the expected asset returns in ((ref)), the pseudo-true value becomes:
where $\frac{O(\tilde{e})}{O(\beta )}\cdot (b^{\prime }b)^{-1}b^{\prime }a$ reflects the difference between the pseudo-true value $\lambda _{F,FM}^{\ast }$ and the generic risk premia $\lambda _{F}.$
The specification of the pseudo-true value ((ref)) therefore shows that it depends on the strength of misspecification, $O( \tilde{e}),$ compared to the identification strength, $O(\beta )$, unless $ (b^{\prime }b)^{-1}b^{\prime }a=0$ so the misspecification error is outside of the space spanned by $\beta $. For example, when we use the specification from the weak instrument/factor literature: $O(\tilde{e})=O(T^{c_{\tilde{e} }})=O_{c_{\tilde{e}}}\times T^{c_{\tilde{e}}},$ $O(\beta )=O(T^{c_{\beta }})=O_{c_{\beta }}\times T^{c_{\beta }},$ with $c_{\tilde{e}}=c_{\beta }=- \frac{1}{2}$ and $O_{c_{\tilde{e}}},$ $O_{c_{\beta }}$ non-zero finite constants:
so depending on $O_{c_{\tilde{e}}}$ being larger or smaller than $ O_{c_{\beta }},$ there can be a considerable difference between $\lambda _{F,FM}^{\ast }$ and $\lambda _{F}.$ Hence, it is important to compare the strength of misspecification reflected by $O(\tilde{e})$ to the strength of identification reflected by $O(\beta )$, in order to gauge the difference between $\lambda _{F,FM}^{\ast }$ and $\lambda _{F}.$
Under correct specification, the pricing error is zero at the true value of risk premia, and remains so when repackaging the assets. The minimizers of the population objective functions underlying different estimators all therefore have the true value of risk premia as their minimizer, and are all invariant to repackaging. Under the standard conditions, these estimators are also consistent for the true value. This is no longer the case in misspecified settings where the minimizers of the population objective functions associated with various estimators (i.e. the pseudo-true values) differ, since there is no longer a value of risk premia at which the population objective function is equal to zero.
Kandel and Stambaugh (1995)\nocite{kanstam95} show that the pseudo-true value for the FM two-pass estimator is not invariant to repackaging of the assets, while the population cross-section generalized least squares (GLS)\ estimator is. The FM two-pass estimator provides a consistent estimator of the pseudo-true value, which makes Kandel and Stambaugh (1995) skeptical about how much of interest the cross-section ordinary least squares (OLS) estimator, or FM two-pass estimator, is under misspecification. They thus show a preference for the cross-section GLS estimator.
\paragraph{Theorem 1.}
For $A$ an invertible $N\times N$ matrix of weights, $A\iota _{N}=\iota _{N}$ with $\iota _{N}$ the $N$-dimensional vector of ones, the FM two-pass pseudo-true value for the orginal test assets, $R_{t},$ does not equal that of the repackaged test assets, $A\times R_{t}.$
Proof: See the Appendix and Kandel and Stambaugh (1995).
Kandel and Stambaugh (1995)'s analysis primarily concerns the population cross-section OLS and GLS objective functions. These population objective functions treat the expected asset returns, $\mu _{R},$ and $\beta $'s as known while we replace them by estimators in the sample objective functions. When extending these population objective functions to sample ones, an important and empirically relevant issue occurs if the proximity of the true $\beta $'s to a reduced rank value is comparable to its estimation error. The resulting risk premia estimators are then exposed to multi-collinearity issues and test statistics, like, for example, the FM\ two-pass $t$-test and its misspecification robust extension by Kan, Robotti, and Shanken (2013), \nocite{krs13} are no longer reliable for conducting tests on the pseudo-true value of the FM two-pass estimator. Test statistics which remain reliable for conducting tests on the pseudo-true value do, however, exist and are based on the continuous updating estimator (CUE) of Hansen, Heaton, and Yaron (1996).\nocite{hhy96} The population objective function of the CUE provides an extension of the GLS population objective function by accounting for the estimation error of all estimable components of the pricing error, $ i.e.$ $\mu _{R}$ and $\beta :$
where $\hat{\mu}_{R}=\bar{R}=\frac{1}{T}\sum_{t=1}^{T}R_{t},$ $\hat{\beta}= \frac{1}{T}\sum_{t=1}^{T}\bar{R}_{t}\bar{F}_{t}^{\prime }\left( \frac{1}{T} \sum_{j=1}^{T}\bar{F}_{j}\bar{F}_{j}^{\prime }\right) ^{-1},$ $\bar{R} _{t}=R_{t}-\bar{R},$ $\bar{F}_{t}=F_{t}-\bar{F},$ $\bar{F}=\frac{1}{T} \sum_{t=1}^{T}F_{t}.$ The specification in the last line of ((ref)) is for a setting of i.i.d. data, so $\Omega =$Var($R_{t}-\beta F_{t})$ and $ Q_{FF}=$Var($F_{t}).$ The GLS population objective function results if we remove the first part of the expression on the bottom line of ((ref) ).
The CUE population objective function normalizes the pricing error, so it is invariant under repackaging and transformations of the factors. The (normalized) sample value of the pricing error used in ((ref)) has unit variance, which implies that, under mild conditions (see, e.g., Shanken (1992)),\nocite{sh92} the sample CUE objective function has a non-central $ \chi ^{2}$ distribution in large samples. The pseudo-true value associated with the CUE then results as
which is also invariant to repackaging.
The population CUE objective function is multi-modal, and the pseudo-true value results from the smallest mode. For example, in an i.i.d. setting, it results from an eigenvalue problem so the number of modes equals the number of eigenvalues/characteristic roots.\footnote{ For the setting of i.i.d. data, a closed-form expression of the pseudo-true value can be provided as resulting from the eigenvector associated with the smallest root of the characteristic polynomial $\left\vert \tau \left(
\right) -\left(
\right) ^{\prime }\Omega ^{-1}\left(
\right) \right\vert =0$ by specifying that eigenvector as $c\binom{1}{ -\lambda _{F,CUE}^{\ast }}$ with $c$ a scalar.} To guarantee that the characteristic root from which the pseudo-true value results, identifies risk premia, the identification strength has to exceed the misspecification strength. These identification properties result because the CUE\ population objective function is the optimized function in the stepwise optimization of a generalized reduced rank objective function, which imposes a reduced rank value on the $N\times (K+1)\ $matrix $\left( \mu _{R} \vdots \beta \right) ,$ see Kleibergen (2007)\nocite{kf04} and Kleibergen and Zhan (2021):\nocite{kz21}
where the expression on the last line is for the setting of i.i.d. data. \footnote{ vec($A)$ is the column vectorization of a matrix $A$ that results from stacking its columns, so vec($A)=(a_{1}^{\prime }\ldots a_{m}^{\prime })^{\prime }$ for $A=(a_{1}\ldots a_{m}).$ tr($A)$ is the trace, or sum of its diagonal elements, of the square matrix $A.$} The second expression in ( (ref)), $Q_{CUE}(l,B),$\ is a normalized distance measure between the $ N\times (K+1)$ matrix $\left( \mu _{R}\text{ }\vdots \text{ }\beta \right) ,$ which is at most of rank $K+1,$ and the $N\times (K+1)$ matrix $B\left( l \text{ }\vdots \text{ }I_{K}\right) ,$ which is at most of rank $K.$ \footnote{ The rank of $B\left( l\text{ }\vdots \text{ }I_{K}\right) $ is at most $K$ because $B$ is an $N\times K$ matrix and $\left( l\text{ }\vdots \text{ } I_{K}\right) $ a $K\times (K+1)$ matrix, so the ranks of both of these are at most $K.$} The sample analog of the CUE population objective function is therefore a rank test on $\left( \mu _{R}\text{ }\vdots \text{ }\beta \right) $. Its minimal value is the $J$-statistic for misspecification.
Since the pseudo-true value of the CUE results from a generic rank test on $ \left( \mu _{R}\text{ }\vdots \text{ }\beta \right) ,$ it does not necessarily represent risk premia when there is misspecification. Consider, for example, a setting where the strength of misspecification, $O(\tilde{e} ), $ is considerably larger than the strength of identification, $O(\beta ),$ which is tiny. The minimization over $(l,B)$ then leads to a tiny value of $ B $ and a very large value of $l$ that does not reflect the generic risk premia. This reasoning is thus also in line with ((ref)) for the FM two-pass approach.
The above point is further illustrated in Figure 1. It shows, for a single factor setting, the contour lines of the pseudo-true values of the FM two-pass estimator and the CUE in deviation from the risk premium in case of correct specification, as a function of the misspecification and identification strengths. Both for FM two-pass (left panel) and CUE (right panel), Figure 1 shows that the pseudo-true values deviate considerably from the baseline risk premium when the misspecification strength exceeds the identification strength. The pseudo-true value of CUE then no longer represents a risk premium, because the closest proximity of $\left( \mu _{R} \text{ }\vdots \text{ }\beta \right) $ from a reduced rank value results mainly from the small value of $\beta $, and much less so from the combination of $\mu _{R}$ and $\beta .$ It is therefore important to be able to diagnose if the pseudo-true value results from such a setting.
\singlespace{ Notes: The pseudo-true values of FM and CUE are defined in ((ref)) and ((ref)), respectively. The baseline risk premium is set to 2, so the contour lines show the deviation of pseudo-true values from 2. The parameters used for Figure 1 are calibrated to the data from Kroencke (2017). The magnitudes of $\tilde{e}$ and $\beta $ vary from 0 to 5.}
\doublespace
To obtain a diagnostic for interpreting the pseudo-true value as a risk premium, we use that the CUE population objective function evaluated at the pseudo-true value equals a rank test on $(\mu _{R}$ $\vdots $ $\beta )$ as shown by ((ref)), which is also the population equivalent of a $J$ -statistic:
It is then always smaller than or equal to an appropriately specified rank test statistic conducted on any sub-matrix of $\left( \mu _{R}\text{ }\vdots \text{ }\beta \right) .$ An example of a rank test on a sub-matrix of $(\mu _{R}$ $\vdots $ $\beta )$ is a rank test on just $\beta ,$ whose sample analog is typically used to test for the identification of the risk premia under correct specification; see, e.g., Cragg and Donald (1997), Kleibergen and Paap (2006) and Robin and Smith (2000).\nocite{cradon97}\nocite{kpaap02} \nocite{robsmit00} This rank test statistic is thus always larger than or equal to the CUE objective function evaluated at the pseudo-true value (i.e. the $J$-statistic in ((ref))), and provides a measure of the identification strength of $\lambda _{F},$ see Kleibergen and Zhan (2021):
When the $J$ misspecification measure is close to the $IS$ identification measure, the pseudo-true value basically results from a close to reduced rank value of $\beta .$ It implies that the pseudo-true value $\lambda _{F,CUE}^{\ast }$ is very large and does not represent risk premia. Hence, only when the $J$ misspecification measure is considerably less than the $IS$ identification strength measure does the pseudo-true value represent risk premia. It shows that when misspecification is present, the identification of risk premia is no longer just reflected by the rank strength value of $ \beta ,$ as is the case under correct specification, but by the difference between a measure of the degree of misspecification and a measure of the rank strength of $\beta $. In case of correct specification, the misspecification measure equals zero, so the identification then solely results from the rank value of $\beta ,$ but not so when misspecification is present. The cut-off for identification is when the measures of misspecification and rank strength of $\beta $ are equal in which case $ \lambda _{F}$ is not identified, while it can be identified as risk premia when the latter exceeds the former.
To summarize, the sample analog of the $IS$ identification measure is a rank test statistic on $\beta $, while the sample analog of the minimal value of the population CUE objective function is the $J$-statistic for misspecification. It is thus important to compare the $IS$-statistic on $ \beta $ with the $J$-statistic for misspecification. When using aligning specifications for the $J$-statistic and the $IS$-statistic on $\beta ,$ the former is always less than or equal to the latter. Close values, however, indicate an issue with identifying risk premia. The estimated values of the risk premia are then also typically very large, which sheds further doubt on whether they can be interpreted as risk premia.\footnote{ It is important to note that we cannot test for the equality of the $J$-test and rank test for $\beta $, since they involve the same estimators. For example, in case $\beta $ is of reduced rank while the expected returns, $ \mu _{R},$ are different from zero, the population values of these two test statistics would be the same, and the joint sampling distribution of their estimators would be degenerate, so we cannot establish a test for their equality.}
To conduct inference on the pseudo-true values of risk premia, it is important to have tests that remain reliable for a wide range of values of the misspecification and identification strengths. We show that this does not hold for the FM $t$-test and its misspecification robust extension by Kan, Robotti, and Shanken (2013), which are size distorted when the identification strength of risk premia is minor. We conduct a small simulation exercise to highlight this sensitivity. Thereafter we state the double robust Lagrange multiplier (DRLM) test from Kleibergen and Zhan (2021), which is size correct for all settings of the misspecification and identification strengths.
We use a single factor setting, so $K=1,$ to show that the FM two-pass $t$ -test and its misspecification robust extension by Kan, Robotti, and Shanken (2013),\nocite{krs13} for conducting inference on the FM two-pass pseudo-true value become unreliable, when the value of the $\beta $-matrix is close to rank deficient. For the single factor setting, a close to rank deficient value of $\beta $ implies that $\beta $ is close to zero. Because of the commonality of misspecification paired with weak identification, we construct the large sample behavior of the FM\ two-pass estimator for a setting where both the $\beta $'s and the misspecification are small. We model this using the weak factor/small $\beta $ and misspecification assumption ((ref)) and ((ref)), and further postulate that both $\beta $ and the misspecification $\mu _{R}-\beta \lambda _{F}$ are drifting to zero at rate $1/\sqrt{T}:$
with $b$ and $a$ $N$-dimensional vectors of constants. The small $\beta $ assumption ((ref)) is common in the weak identification literature; see, e.g., Staiger and Stock (1997),\nocite{stst97} Kleibergen (2005, 2009) and Kleibergen and Zhan (2018, 2021).\nocite {kleibergen2018identification}\nocite{kf09} \nocite{kf00a} \nocite{kz20} It leads to the small values of the $F$-statistic testing the joint significance of the $\beta $'s that we often observe; see Kleibergen and Zhan (2015).\nocite{kz15} The small misspecification assumption further accommodates the small but significant values of the $J$-statistic that are regularly seen.
\paragraph{Theorem 2:}
For i.i.d. data and under the weak $\beta $ and misspecification assumption ( (ref)), the large sample distribution of the FM two-pass estimator, $\hat{\lambda}_{F}=(\hat{\beta}^{\prime }\hat{\beta})^{-1}\hat{ \beta}^{\prime }\hat{\mu}_{R}$, consists of four components:
with $e=a-b(b^{\prime }b)^{-1}b^{\prime }a$; $\psi _{\mu }$ and $\psi _{\beta }$ are independent normally distributed $N$-dimensional random vectors with mean zero and covariance matrices var($R_{t})=\Omega +\beta Q\beta ^{\prime }$ and $\Omega Q^{-1},$ $\Omega =$Var($R_{t}-\beta F_{t})$ and $Q_{FF}=$Var($F_{t}).$
Proof: See the Appendix.
The four different components of the large sample behavior of the FM two-pass estimator in Theorem 2 are characterized by:
The third and fourth components lead to a large sample distribution for the FM two-pass estimator which is not a normal one for the empirically relevant setting of small values of $b.$ Test statistics based on the FM two-pass estimator, like the FM $t$-test and the KRS $t$-test, therefore become size distorted for such small values. To illustrate this, Figure 2 shows the simulated rejection frequencies of 5% significance FM and KRS $t$-tests on $ \lambda _{F,FM}^{\ast }$ for a range of values of the misspecification and identification strengths.\footnote{ Throughout the paper, simulated data are generated from the linear factor model using parameters calibrated to the data from Kroencke (2017).} Figure 2.1 shows the rejection frequencies for H$_{0}:\lambda _{F,FM}^{\ast }=0$, and Figure 2.2 is for $\lambda _{F,FM}^{\ast }$ corresponding with the values resulting from Figure 1.
Figure 2.1 shows that the FM $t$-test and the KRS $t$-test are conservative or correctly sized when testing H$_{0}:\lambda _{F,FM}^{\ast }=0$ and there is no misspecification so $\tilde{e}=0.$ For increasing values of $\tilde{e}$ , however, the FM $t$-test and the KRS $t$-test over-reject for small values of $\beta $, i.e. the rejection frequencies are larger than the nominal 5%. For the FM $t$-test, this over-rejection also extends to larger values of $ \beta $.
For testing H$_{0}:\lambda _{F,FM}^{\ast }$ corresponding with the pseudo-true values resulting from Figure 1, Figure 2.2 shows that the FM $t$ -test and the KRS $t$-test severely over-reject for small values of $\beta $ which are less than $\tilde{e}$. When $\beta$ approaches zero, the rejection frequencies are even equal to one, which is in line with Kan and Zhang (1999).\nocite{kz99}
The decomposition in Theorem 2 can also be conducted for the GLS risk premia estimator, so test statistics based on it, such as the GLS $t$-test, become similarly size distorted. We therefore use a statistic based on the CUE: the DRLM statistic proposed in Kleibergen and Zhan (2021), whose asymptotic distribution is bounded by the $\chi _{K}^{2}$ distribution. When using appropriate critical values from the $\chi _{K}^{2}$ distribution, the DRLM\ test of hypotheses specified on the pseudo-true value of the CUE remains size correct for general levels of misspecification and identification. In the Appendix, we state the DRLM statistic, and provide a brief discussion of its implementation.
Figure 3 presents the rejection frequencies of a 5% significance DRLM\ test of H$_{0}:\lambda _{F,CUE}^{\ast }=0$. We note that, unlike the large sample distributions of the FM and KRS $t$-tests, the limiting distributions of the statistics underlying the DRLM test do not depend on the tested parameter or the covariance matrices $\Omega $ and $Q_{FF}$. In contrast with Figure 2 for the simulated sizes of the FM and KRS $t$-tests, Figure 3 shows that the DRLM test is size correct (i.e. the rejection frequencies do not exceed the nominal 5%), and is conservative for combined small identification and misspecification strengths.
We briefly illustrate the power of the FM and KRS $t$-tests compared to the DRLM test. A more extensive power study of the DRLM test is conducted in Kleibergen and Zhan (2021). We first show the power for a correct specification, so all the aforementioned tests target the same value of the risk premium, while there is also strong identification. Thus, we have the ideal setting that $\beta $ is sizeable while $\tilde{e}=0$ in the data generation process. All the examined tests are therefore size correct in Figure 4a-b at the hypothesized value, i.e. all the rejection frequencies are near the nominal level of 5% when the distance to the tested risk premium is zero. Since there is no misspecification, the power curves of FM $ t$ (dotted black) and KRS $t$ (dashed green) largely overlap in Figure 4a. The comparison of Figure 4a and 4b, however, also shows that the DRLM test can have more power than FM and KRS $t$-tests. This results since the DRLM test is based on the GLS framework, while the FM and KRS $t$-tests for Figure 4 are based on OLS.
In contrast with Figure 4a-b, we allow for misspecification as well as weak identification in Figure 4c-d. This is achieved by considering smallish $ \beta $ and nonzero $\tilde{e}$ in the data generation process. For this scenario, Figure 4c shows that the FM and KRS $t$-tests are no longer size correct, while their power curves substantially differ from those in Figure 4a. In contrast, the DRLM test, which involves the power improvement rule mentioned in the Appendix, remains size correct in Figure 4d.
Overall, Figure 4 illustrates that it is appealing to use the DRLM test for conducting inference on risk premia. In contrast, the conventional FM $t$ -test, together with the KRS $t$-test, is jeopardized by joint misspecification and weak identification, both of which are prevalent in empirical studies as we show next.
Before we turn to empirical applications of the tests on the risk premia, we first apply the diagnostic statistics which help to gauge whether we can interpret the risk premia accordingly. The $IS$-statistic, which tests for a reduced rank value of $\beta ,$ is commonly used to test for identification of the risk premia in correctly specified settings. In misspecified settings, it is, however, no longer just the $IS$-statistic which governs the identification and interpretation of the pseudo-true value of the risk premia, but its difference with the $J$-statistic, which equals the minimal value of the sample CUE objective function. This is, for example, shown by the deviation of the FM two-pass pseudo-true value in ((ref)) from its counterpart under correct specification, and further illustrated in Figure 1. The $J$ and $IS$ statistics have well established limiting distributions under their hypotheses of interest, which are $\chi _{N-K}^{2}$ for the $J$-statistic under correct specification, and $\chi _{N-K+1}^{2}$ for the $IS$-statistic under a reduced rank $\beta $ matrix; see the Appendix for their explicit expressions. We are, however, mainly interested in these statistics because of the inequality between them, $J$ -statistic $\leq $ $IS$-statistic, and because a close proximity between these two statistics shows that the pseudo-true value does not identify risk premia as discussed in Section 2. We therefore compute these two statistics first for eight well known specifications of the linear asset pricing model, and second for the specifications resulting from the factor zoo of Feng, Giglio, and Xiu (2020).\nocite{feng2020taming}
Figure 5 shows a scatter plot of the $J$ and $IS$ statistics for eight well known specifications of the linear asset pricing model: Fama and French (1993), Jagannathan and Wang (1996), Yogo (2006), Lettau and Ludvigson (2001), Savov (2011), Adrian, Etula, and Muir (2014), Kroencke (2017) and He, Kelly, and Manela (2017).\footnote{We thank the authors of Jagannathan and Wang (1996), Yogo (2006), Lettau and Ludvigson (2001), Savov (2011), and Kroencke (2017) for sharing their data. For the models of Fama and French (1993), Adrian, Etula, and Muir (2014), and He, Kelly, and Manela (2017), we use the extended data of risk factors and test assets as in Lettau, Ludvigson, and Ma (2019).\nocite{llm19}} In line with common practice, we incorporate the zero-$\beta $ return, $\lambda _{0},$ while the factors used in the eight different specifications are:
\doublespace
The plotted points in Figure 5 roughly exhibit two patterns, which we illustrate by using different colors (red and blue). For Fama and French (1993), Jagannathan and Wang (1996), Savov (2011) and Kroencke (2017), we observe both large $J$ and $IS$ statistics, so the corresponding models appear to be misspecified. In contrast, for Lettau and Ludvigson (2001), Yogo (2006), Adrian, Etula, and Muir (2014), and He, Kelly, and Manela (2017), we encounter small $J$ and $IS$ statistics, so these models are likely to be weakly identified.
The $IS$-statistics for the first set of four specifications are large and mostly significant but they are also close to their respective $J$ -statistics, which is revealed by the proximity of these points to the 45-degree line. We can therefore not conclude from these $IS$-statistics that we identify risk premia. The most likely setting for identification is for Fama and French (1993), whose ($IS$, $J$) point is most distant from the 45-degree line.
For the second set of four points, the $J$-statistics are small and mostly insignificant, indicating that the models might not be misspecified. However, their corresponding $IS$-statistics are also similarly small, which implies that the low values of the $J$-statistics are induced by the low values of the $IS$-statistics, since $J$-statistics are necessarily exceeded by $IS$-statistics. Hence, we cannot identify risk premia for these specifications.
Panel A in Table (ref) states the values of the $J$ and $IS$ statistics plotted in Figure 5. Panel B states these statistics when the zero-$\beta $ return is removed, so $\lambda _{0}=0.$ All statistics in Panel B therefore exceed their corresponding counterparts in Panel A. Removing the zero-$\beta $ return thus increases both the misspecification and identification. The misspecification has increased since we removed a parameter from the pricing equation, while identification has improved since removing the zero-$\beta $ return allows to identify risk premia when the $ \beta $'s are constant over the assets, which was not so when the zero-$ \beta $ return was included. For some of the specifications, the increase of the $J$ and $IS$ statistics is disproportional. This is most notably so for the Fama and French (1993) specification, where the $IS$-statistic increases ninefold while the $J$-statistic does so only minorly. For the other specifications, the increase of the $IS$-statistic typically exceeds that of the $J$-statistic, but not by an amount which makes it clear that the risk premia become well identified as is the case for the Fama and French (1993) specification. For more than half of the specifications, the removal of the zero-$\beta $ return has therefore little effect on the identification of risk premia.
We note that the $J$ and $IS$ statistics presented in Figure 5 are just for illustrative purposes. We do not aim to use these statistics to strictly reject or favor any model in Figure 5, since there are many other issues involved. These issues include, for example, that the models used for Figure 5 contain various numbers of risk factors; in addition, their corresponding empirical studies have used non-identical test assets; furthermore, the considered time periods and frequencies also vary to a large extent. To address these concerns, we next use the factor zoo data from Feng, Giglio, and Xiu (2020) to investigate misspecification and weak identification in a more systematic manner.
How prevalent are misspecification and weak identification in empirical asset pricing studies? With this question in mind, we extend our analysis to the factor zoo collected by Feng, Giglio, and Xiu (2020),\nocite {feng2020taming} which covers one hundred and fifty risk factors from July 1976 to December 2017, so $T$ = 498. We use the twenty-five Fama and French size and book-to-market portfolios as test assets, which have been widely used as the default choice.
In line with existing studies, we evaluate all possible specifications of linear factor models resulting from the factor zoo with $K=1$, $2$, $3$, $4$ , $5$ and $6$ factors. For $K=1$, we therefore consider $C_{150}^{1}=150$ single factor models. Similarly, we examine $C_{150}^{2}=11,175$ $ (=150\times 149/2)$ two-factor models; $C_{150}^{3}=551,300$ three-factor models; $C_{150}^{4}=20,260,275$ four-factor models; $ C_{150}^{5}=591,600,030 $ five-factor models; and $ C_{150}^{6}=14,297,000,725 $ six-factor models.\footnote{ We used high-performance computers to conduct this study.} For each model, we compute its $J$ and $IS$ statistics while incorporating the zero-$\beta $ return.
Table (ref) states the frequencies of, at the 5% level, significant values of the $J$-statistics, and insignificant values of the $IS$ -statistics, signaling misspecification and weak identification, respectively. Table (ref) shows that, when we increase the number of factors, the resulting factor models tend to be less misspecified, while becoming weaker identified. This results naturally since on the one hand, adding more factors helps to better explain asset returns, so the resulting models are less likely to be rendered misspecified; while on the other hand, since some factors could be closely related to others, including more factors decreases the distance of the $\beta $ matrix from a reduced rank value, which leads to weaker identified models. We note that the reported percentages in Table (ref) likely understate the severeness of misspecification, because the misspecification $J$-statistic tends to be insignificant under weak identification, as we have discussed for Figure 5.
Overall, Table (ref) shows that the majority of the examined models seem to suffer from misspecification and/or weak identification. Apparently, there is also a trade-off between the reported percentages of misspecification and weak identification in Table (ref). We previously have, however, shown that we cannot analyze the $J$ and $IS$ statistics in isolation, as in Table (ref), to determine whether risk premia are identified, so we next analyze them jointly.
Because the $J$ and $IS$ statistics jointly indicate if the risk premia are identified, the six pictures in Figure 6 show the bivariate empirical density functions (histograms) of the $J$ and $IS$ statistics resulting from all the specifications having one to six factors presented in Table 2. Figure 7 shows the contour lines of these six empirical density functions.
When $K=1$, Figures 6a and 7a show that almost all single factor models are associated with large significant (at the 5% level) $J$-statistics. This is as expected, since it is unlikely that a single factor explains all the variation in the cross-section of expected asset returns. Specifically, 148 out of 150 single factor models are deemed misspecified by their significant $J$-statistics at the 5% level, leading to $148/150\approx 98.67\%$ in Table (ref). The remaining two single factor models have small $J$ -statistics (see the bottom left of Figure 6a), but their $IS$-statistics are also small. One of these two $IS$-statistics is even insignificant at the 5% level, leading to the $1/150\approx 0.67\%$ for weak identification in Table (ref), while the other is insignificant at the 1% level. Thus, these two factors are of poor quality, and their resulting single factor models should also be deemed misspecified. The $J$-statistic fails to signal misspecification for these two models, because the small $IS$ -statistic forces it to be very small, given that the $J$-statistic is less than or equal to the $IS$-statistic. Joint misspecification and weak identification therefore occur in these two single factor models.
When we increase the number of factors so $K=2$, $3$, $4$, $5$ and $6,$ Figures 6b-f and 7b-f show two clear patterns:
While Table (ref) shows that for $K=6,$ more than 50% of the examined specifications are weakly identified, the second pattern listed above implies that many more specifications are weakly identified, and similarly so for specifications involving fewer factors. This is analogous to what we observed for Figure 5. Taken all together, it all shows that joint misspecification and weak identification is a common problem that needs to be addressed.
Encouragingly, Figures 6 and 7 also show that there are specifications for which the ($IS$ , $J$) combination is distant from the 45-degree line. For these specifications, the $IS$-statistics exceed their $J$-statistics to a larger extent when compared to those in Figure 5. From a purely statistical point of view, the specifications with $J$ much smaller than $IS$ are worth investigation, since their pricing errors tend to be small while the risk premia are likely to be well identified. It would be interesting if researchers could further relate these models to economic theories. Yet given the large number of such models as shown by Figure 6, we leave them for future research.
We show the importance of $J,$ $IS$ and DRLM statistics for applied asset pricing by revisiting two prominent examples: the Fama and French (1993) model and the conditional consumption capital asset pricing model from Lettau and Ludvigson (2001). Table (ref) therefore reports the estimation results for the three-factor specification used in Lettau and Ludvigson (2001), and for Fama and French (1993) using three different data sets. Accordingly, Figure 8 shows the joint 95% confidence sets resulting from DRLM\ for all four specifications in Table (ref).
The Fama and French (1993) three-factor model has been widely used as a benchmark in the asset pricing literature; see, e.g., Lettau and Ludvigson (2001) and Lettau, Ludvigson, and Ma (2019). In line with the large, significant (at the 1% level) $J$-statistics in Table (ref), the Fama and French (1993) three-factor model is well acknowledged to be misspecified. Yet this model appears able to explain the cross-section of asset returns, as reflected by the reported large cross-sectional $R^{2}$.
Using quarterly data from Lettau and Ludvigson (2001) and Lettau, Ludvigson, and Ma (2019), the two-pass procedure of Fama and MacBeth (1973) suggests that the risk premia on SMB and HML are both positive; in contrast, the point estimate of the market premium is negative in Lettau, Ludvigson, and Ma (2019), but positive in Lettau and Ludvigson (2001); see Panel A versus Panel B in Table (ref). Because Lettau, Ludvigson, and Ma (2019) use longer time series than Lettau and Ludvigson (2001), its $IS$-statistic is considerably larger (106.81 vs. 47.01), reflecting that more information is available for identification in Lettau, Ludvigson, and Ma (2019). On the other hand, the larger $J$-statistic for Lettau, Ludvigson, and Ma (2019) (59.34 vs. 45.38) implies more severe misspecification. A question thus arises: how do we reconcile the seemingly conflicting findings in Lettau, Ludvigson, and Ma (2019) and Lettau and Ludvigson (2001) for $R_{m}$ in the Fama and French (1993) three-factor model?
\doublespace
To help answer this question, we employ the DRLM test, since this test is robust to misspecification as well as the strength of identification. Specifically, we test every risk premium value between -5 and 5, since Table (ref) Panel A and Panel B indicate that risk premia on all three factors likely lie within this range. The resulting joint confidence sets at the 5% level are presented in Figure 8a-b, using the data from Lettau, Ludvigson, and Ma (2019) and Lettau and Ludvigson (2001), respectively.
Interestingly, Figure 8a-b show that the risk premia on SMB and HML are strongly identified, while the risk premium on $R_{m}$ is not. These findings remain similar no matter whether we adopt the data from Lettau, Ludvigson, and Ma (2019) or Lettau and Ludvigson (2001). In both Figure 8a-b, we can not reject any risk premium value on $R_{m}$ between -5 and 5, which thus helps reconcile the difference in the confidence intervals for the risk premium on $R_{m}$ reported in Table (ref) Panel A and Panel B.
For SMB and HML, however, the DRLM test yields tight 95% confidence sets for their risk premia in Figure 8. For example, Figure 8a implies that the range of the risk premium on SMB is (0.59, 1.06), while the range of the risk premium on HML is (0.04, 2.00). Similar ranges can be derived from Figure 8b. All these sets thus largely overlap with those resulting from the FM $t$-test as reported in Table (ref) Panel A and Panel B.
One might wonder what causes the large difference in the risk premia between $R_{m}$ and SMB, HML. To illustrate, we present their $\beta $ estimates and associated $t$-statistics in Table (ref). It is clear in Table (ref) that the three Fama and French (1993) factors are all closely related to the test asset returns as reflected by the significant $t$-statistics. There is, however, little cross-sectional variation in the estimated betas of $R_{m}$, i.e. the $\beta $ estimates on $R_{m}$ are all close to 1. Thus, if the zero-$\beta $ return is incorporated, we have near-multicollinearity in the $(\iota _{N}$ $\vdots $ $\beta )$ matrix for the cross-sectional regression, causing the market risk premium to be weakly identified. Consequently, we observe in Table (ref) Panels A and B the seemingly conflicting risk premium values on $R_{m}$, but not for SMB and HML. For the same reason, we observe in Figure 8a-b the wide range for the risk premium on $R_{m}$, but narrower ones for SMB, HML.
To compare with Fama and French (1993), we consider the conditional consumption capital asset pricing model from Lettau and Ludvigson (2001) where the three factors are consumption growth, $\triangle c$, consumption-wealth ratio, $cay$, and their interaction, $\triangle c\times cay$. The significant FM $t$ and Shanken $t$-statistics on the interaction $ \triangle c\times cay$, the relatively small, insignificant (at the 5% level) $J$-statistic, together with the large cross-sectional $R^{2}$ reported in Panel C of Table (ref) provide considerable motivation for this model for asset pricing.
The small $IS$-statistic in Panel C of Table (ref), however, shows that the three-factor model of Lettau and Ludvigson (2001) is just weakly identified. Furthermore, given its proximity to the $J$-statistic, the small value of the $J$-statistic also results from it, since the $J$ -statistic is at most equal to the $IS$-statistic. The risk premia can therefore not be identified. Consequently, the significant FM $t$-statistic on $\triangle c\times cay$ should be interpreted with caution, since the FM $ t$-test is now unreliable and this reasoning similarly applies to the KRS $t$ -test. Also, as warned by Kleibergen and Zhan (2015)\nocite{kz15}, weakly identified models can yield spuriously large cross-sectional $R^{2}$'s, which should then not be taken as the evidence in support of asset pricing.
Next, we apply the DRLM test to the three-factor model of Lettau and Ludvigson (2001). As shown by Figure 8c, we can not reject any hypothesized value for risk premia in the range of [-5, 5], reflecting that the risk premia can not be identified for the conditional consumption capital asset pricing model in Lettau and Ludvigson (2001), which is in line with the proximity of the $J$ and $IS$ statistics.
The challenge to identify the risk premia in Lettau and Ludvigson (2001) is further explained by Table (ref), which contains the $\beta $ estimates and their associated $t$-statistics. Table (ref) shows that $\triangle c\times cay$ is poorly correlated with asset returns, which makes a column of $\beta $ statistically close to zero (i.e. tiny $t$-statistics in the last column). Thus, the full rank condition of $\beta $ is problematic, which leads to the weak identification problem in Lettau and Ludvigson (2001). Consequently, we observe a small $IS$-statistic, which tests for a full rank value of $\beta ,$ in Panel C of Table (ref), and uninformative confidence sets in Figure 8c.
In light of the findings in Figure 8a-c, one might wonder when the DRLM test could yield an informative confidence set for the risk premia. We show that a bounded confidence set of the risk premia is feasible if we just have more time series observations, or if we remove the zero-$\beta $ return, so $ \lambda _{0}=0,$ as presented in the next two subsections, respectively.
With more time series observations, we expect more information in the data and consequently, stronger identification of the risk premia. Figure 8d therefore uses monthly data from July 1963 to June 2021, so $T=696,$ for the Fama and French (1993) three factors and the twenty-five size and book-to-market sorted portfolios, which are downloaded from French's online data library. Table (ref) (right panel) shows the resulting $\beta $ estimates from which it is clear that the standard errors of the estimates have almost halved, which doubles the $t$-statistics testing their significance, while there is also more variation in the $\beta $ estimates for $R_{m}$ when compared to the left panel of Table (ref) with fewer observations. The estimate of $\beta $ is consequently distant from a reduced rank value as shown by the large $IS$-statistic, 425.55, in Panel D of Table (ref). It considerably exceeds the resulting $J$ -statistic, 52.25, indicating strong identification of the risk premia.
In line with such strong identification, Figure 8d shows that the joint 95% confidence set of the risk premia on the Fama and French (1993) factors resulting from the DRLM\ test lies in a tight region of the 3-dimensional space. Specifically, this confidence set contains all values of the risk premia that are not rejected by the DRLM test at the 5% level. Combining all these values results in the red joint confidence set plotted in Figure 8d. Moreover, projecting the joint confidence set to each axis yields the 95% confidence intervals for each risk premium: (-1.22, 0.18) for $R_{m}$, (0.12, 0.26) for SMB, and (0.24, 0.41) for HML, respectively. These 95% confidence intervals are also comparable to, but often smaller than those reported in Panel D of Table (ref) by inverting the FM $t$, Shanken $t$ and KRS $t$ tests.
Instead of using richer data, we can also remove the zero-$\beta $ return, so impose $\lambda _{0}=0,$ to improve the identification of the risk premia. When we do so by using the data from Lettau, Ludvigson, and Ma (2019), the $J$-statistic equals 87.47 and the $IS$-statistic becomes 974.39. Hence, the $IS$-statistic has increased ninefold while the $J$ -statistic remains almost the same compared with the specification including the zero-$\beta $ return in Panel A of Table (ref). It indicates that the risk premia are now well identified.
To illustrate further, we use DRLM to construct the joint 95% confidence set in Figure 9, where $\lambda _{0}=0$ is imposed for the Fama and French (1993) model. Figure 9 is to be compared with Figure 8a. They use the same data, but $\lambda _{0}=0$ is imposed for Figure 9 but not for Figure 8a. Clearly, the $\lambda _{0}=0$ restriction strongly improves the identification of the risk premia, so we observe a bounded 95% confidence set in Figure 9. Projecting the joint confidence set to each axis leads to 95% confidence intervals for the risk premia on $R_{m}$, SMB, and HML, respectively: $(1.52,1.84)$ for $R_{m}$, $(0.66,1.06)$ for SMB, and $ (0.59,1.93)$ for HML.
Since the three factors are traded, we can also infer the 95%\ confidence sets on their risk premia by using the average value of the respective factor $\pm $ 1.96 $\times $ standard deviation, which leads to $(0.41,2.80)$ for $R_{m}$, $(0.18,1.78)$ for SMB and $(0.05,1.81)$ for HML. These confidence intervals are comparable to those derived above by the projection method, but are considerably wider. All these empirical findings therefore lend credibility to the proposed DRLM test.
\doublespace
Though misspecified, the Fama and French (1993) three-factor model has been well documented to exhibit relatively strong correlations with asset returns. Indeed, the model leads to large $IS$-statistics as we report in Table (ref). Astonishingly, even when the $IS$-statistic is significant and exceeds 100 (see Panel A of Table (ref)), we still find that the risk premium on $R_{m}$ is only weakly identified in the beta representation. Given that a large number of empirical studies, as indicated by Table (ref), have weaker strength of identification than Fama and French (1993), it is therefore unlikely that their risk premia could be precisely identified. To achieve stronger identification, we therefore need to adopt richer data or impose extra restrictions, as indicated by Figure 8d and Figure 9, respectively.
The conditional consumption capital asset pricing model in Lettau and Ludvigson (2001) serves as an example where the FM $t$-statistic, the misspecification $J$-statistic, and the cross-sectional $R^{2}$ line up nicely to show support for the model. Yet the model appears to lack identification, as reflected by the tiny difference between the $J$ and $IS$ statistics, and the wide confidence set resulting from the DRLM test, since the rank condition of $\beta $ is likely violated. These empirical findings thus clearly show the importance of adopting the $J,$ $IS$ statistics and the DRLM test for assessing asset pricing models.
In the Appendix, we further extend our empirical analysis to the $q$-factor model of Hou, Xue, and Zhang (2015). \nocite{hou2015digesting} In line with Figure 9, we find that the DRLM test leads to bounded 95% confidence sets for the risk premia on all four factors in the $q$-factor model. In particular, for the first three factors (R_MKT, market excess returns; R_ME, size factor; R_IA, investment factor), we find that the 95% confidence sets resulting from DRLM are significant at the 5% level so they exclude zero. Yet for the fourth factor (R_ROE, equity factor), the confidence set is bounded but relatively wide. These findings are consistent with those in Kleibergen and Zhan (2015), who show that the commonly used 25 Fama-French portfolios have a strong factor structure, i.e., the majority of variations in these portfolios are likely captured by three factors, so it is hard to precisely identify risk premia on all four factors in the $q$ -factor model by using such test assets in the two-pass approach.
An alarmingly large number of factors in the asset pricing literature are able to yield significant $t$-statistics on their risk premia, together with seemingly promising misspecification $J$-statistics and cross-sectional $ R^{2}$'s. The credibility of these conventional statistics, however, is threatened by misspecification and weak identification, both of which are prevalent as we document in this paper. We show that failure to account for both misspecification and weak identification could easily lead to erroneous conclusions. To remedy these problems, we suggest that the $J$-statistic for misspecification, the $IS$-statistic for identification strength, and the DRLM test for risk premia become part of a toolkit that helps to provide trustworthy diagnosis and inference in future studies.
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