Tsay Ch9 - Principal Component Analysis and Factor Models

Dimension reduction is essential to search for the underlying structure of the assets - called factors.

Three types of factor models -
1) Macroeconomic factor models - GDP growth, interest rates, inflation, unemployment - observable factors using regression
2) Fundamental factor models - firm size, book and market values, industrial classification.
3) Statistical factor models - non-observable  or latent variables e.g. PCA

General Factor Model

For $m$ factors, $k$ assets, and $T$ time periods let $r_{it}$ be the return of asset i at time period t. The factor model is

$$\pmb{r}_{t}=\pmb{\alpha}+\pmb{\beta}\pmb{f}_t+\pmb{\epsilon}_{t}, \qquad t = 1,...,T$$
where $\pmb{\beta}$ is a $k\times m$ factor loading matrix and $\pmb{\epsilon}_t$ is the error vector with $Cov{\pmb{\epsilon}_t}=\pmb{D}=diag[\sigma^2_1,...,\sigma^2_k]$, a $k\times k$ diagonal matrix. The covariance matrix of the returns $\pmb{r}_t$ is then given by:
$$Cov(\pmb{r}_t)=\pmb{\beta}\pmb{\Sigma}_f\pmb{\beta}^T+\pmb{D}$$

Macroeconomic factor models

Macroeconomic factors are observable. We can convert the general factor model into Multiple Linear regression setup and estimate the factors. This estimation does not impose the constraint of $\epsilon_{it}$ being uncorrelated, so may not be efficient in general. The best known single factor model is the market model (Sharpe 1970). The $R^2$ can reach up to 50%, showing the significance of common market factor. One simple trick to compare factor based covariance matrix with sample covariance matrix is to use the global minimum variance portfolio (GMVP). For a given covariance matrix $\Sigma$, the GMVP $\omega$ solves $min\sigma^2_{p,\omega}=\omega^T\Sigma\omega$, such that $\omega^T\pmb{1}=1$ given by

$$\omega=\frac{\Sigma^{-1}\pmb{1}}{\pmb{1}^T\Sigma^{-1}\pmb{1}}.$$
It is also important to verify that the residual covariance matrices do not have large off-diagonal elements, to fit the factor model criteria.

Ross (1986) considers multi-factor model consisting of unexpected changes or surprises (e.g. residuals after fitting VAR(3) model to seasonally adjusted CPI and unemployment growth numbers). The explanatory power is low.

Fundamental factor models

BARRA factor method treats the observed asset specific fundamentals as the factor betas $\beta_i$, and estimates the factor $f_t$ at each time index $t$ via regression. Fama and French construct their factors based on hedge portfolio which depend on the fundamentals. For BARRA factor model $$\widetilde{\pmb{r}}_t = \pmb{\beta} \pmb{f}_t+\pmb{\epsilon}_t,$$ where $\widetilde{\pmb{r}_t}$ is the mean-corrected returns. We need WLS setup since the regression is not homogeneous, the estimate would be $$\hat{\pmb{f}_t}=(\beta D^{-1}\beta^T)^{-1}(\beta D^{-1}\beta^T\widetilde{r_t}).$$ We estimate the diagonal covariance matrix of errors from OLS first and then use it to estimate the factors using WLS equation. Cross-correlations in errors are ignored. The diagonal covariance matrix of final errors $\hat{D_g}$ and the covariance matrix of estimated factor realizations $\hat{\Sigma}$ can be used to derive the covariance matrix of the original returns as $$Cov(r_t)=\beta\hat{\Sigma}_f\beta^T+\hat{D_g}.$$ In practice, the sample mean or returns are not different from zero, so one may not need to remove the sample mean before fitting the BARRA factor model.

Fama-French approach used a two-step procedure. First, they sorted the assets based on the value of three fundamentals (market excess returns, small vs big cap, value vs growth stocks). They formed the hedge portfolio which is long top quintile and short the bottom quintile. The observed return on this hedge portfolio is the factor realization for the given asset. Finally, given the factor realizations calculate betas using regression.

Principal component analysis

We look to find linear combinations which explain the most variance and are orthogonal to each other, with weights summing to one. This is done on covariance or correlation matrix which are non-negative definite and hence have spectral decomposition. For covariance matrix the variance explained is $\lambda_i/\sum \lambda$, which becomes $\lambda_i/k$ for a correlation matrix, since $Tr(\rho_r)=k$.

Statistical factor analysis

The aim is to identify a few factors that can account for most of the variations in the covariance or correlation matrix of the data. The assumption of no serial correlations is all right for low frequency data but not accurate for higher frequencies. Serial correlations should first be removed parametrically. We then construct orthogonal factor model. Since both the loadings and the factors are unobservable it is different from other factor models. For the Statistical factor model $r_t - \mu = \beta f_t + \epsilon_t$, we have the assumptions $E[f_t]=0$, $Cov[f_t]=\pmb{I}_m$, $E[\epsilon_t]=0$, $Cov[\epsilon_t]=D=diag(\sigma^2_1,...,\sigma^2_k)$ and $E[f_t\epsilon^T]=0$. These are not uniquely determined. This can be estimated either using Principal Component Method or Maximum Likelihood estimation, with specified $k$. Factor rotation can be used for interpretation using varimax criteria.

Left out sections: 9.6