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Does a factor model always include traded portfolios or can it also be other variables? For example, a four factor model of

$R^e_{it}=\alpha_i+\beta^{M}_{i}\text{MKT}_t+\beta^{SMB}_{i}\text{SMB}_t+\beta^{HML}_{i}\text{HML}_t+\beta^{CPI}\text{CPI}_t$, where we have the typical Fama-French three factor model augmented with values of Consumer Price Index (So not a traded portfolio). Can we test individual (5x5 size/value) portfolio $\alpha s$ for such a model or does it not work this way?

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A famous (APT) factor pricing model was developed by Chen, Ross and Roll (1986). The model suggests $$R_{i,t}=a_i + \beta_{i,MP}MP_t + \beta_{i,UI}UI_t + \beta_{i,DEI}DEI_t+\beta_{i,UPR}UPR_t+\beta_{i,UTS}UTS_t+\varepsilon_{i,t},$$ where $\varepsilon_{i,t}$ is an idiosyncratic term. The factors are not all tradable returns. $UPR$ and $UTS$ are, but $MP$, $UI$ and $DEI$ are not. Nonetheless, you can have these factors in your model and use, for example, Fama and MacBeth (1973) regressions to estimate the risk premiums. The individual factors are

  • Monthly growth in industrial production $$MP_t = \ln\left(\frac{IP_t}{IP_{t-1}}\right)$$

  • Unexpected inflation $$UI_t=I_t-\mathbb{E}_{t-1}[I_t]$$

  • Change in expected inflation $$DEI_t=\mathbb{E}_t[I_{t+1}]-\mathbb{E}_{t-1}[I_t]$$

  • Default risk premium $$UPR_t=\text{Baa or under bond returns}_t - \text{Long-term government bond returns}_t$$

  • Term structure premium $$UTS_t= \text{Long-term government bond returns}_t - \text{Treasuary-bill rate}_{t-1}$$

Sometimes, a researcher may want to use tradable portfolios. Chan, Karceski, and Lakonishok (1998, JFQA) or Cooper and Priestley (2011, JFE) are examples who follow this approach and describe how to obtain mimicking portfolios for the CRR factors.

In doubt, it's always good to see what John Cochrane has to say.

The pricing implications of any model can be equivalently represented by its factor-mimicking portfolio. If there is any measurement error in a set of economic variables driving $m$, the factor-mimicking portfolios for the true $m$ will price assets better than an estimate of $m$ that uses the measured macroeconomic variables.

Thus, it is probably not a good idea to evaluate economically interesting models with statistical horse races against models that use portfolio returns as factors. Economically interesting models, even if true and perfectly measured, will just equal the performance of their own factor-mimicking portfolios, even in large samples. Add any measurement error, and the economic model will underperform its own factor-mimicking portfolios.

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