In dppalomar/covFactorModel: Covariance Matrix Estimation via Factor Models

Explanation of the method

The factor model decompose the stock returns into two parts: low-dimensional factors and idiosyncratic residual noise. There are three basic type of factor models [@tsay2005analysis], namely, macroeconomic, fundamental and statistical. Assuming there are $N$ stocks in market and we have $T$ observation, then all three type of factor models can be expressed using linear form: $$x_{i,t} = \alpha_{i} + \beta_{1,i}f_{1,t} + \dots + \beta_{K,i}f_{K,t} + \epsilon_{i,t}, \; t = 1, \dots, T$$ where $K$ is the number of factors, $\alpha_{i}$ is intercept of stock $i$, $\mathbf{f}{k} = [f{k,1}, \dots, f_{k,T}]^{T}$ is common factors $k$, $\boldsymbol{\beta}{i} = [\beta{1,i}, \dots, \beta_{K,i}]^{T}$ is factor loading of stock $i$ and $\epsilon_{i,t}$ is residual term for stock $i$ at time $t$. With compact combination $\mathbf{F}=\left[\begin{array}{ccc} \mathbf{f}{1} & \cdots & \mathbf{f}{K}\end{array}\right]$ and $\mathbf{x}{i} = [x{i,1}, \dots, x_{i,T}]$, it can also be written into vector form: $$\mathbf{x}{i} = \mathbf{1}{T} + \mathbf{F} \boldsymbol{\beta}{i} + \boldsymbol{\epsilon}{i}, \; i=1,\dots,N$$

`factorModel()`: Build factor model for given data

The goal of factorModel() is the decomposition of a $T\times N$ data matrix $\mathbf{X}$ into factors and residual idiosyncratic component. User can choose different types of factor models, namely, macroeconomic, BARRA (a special case of fundamental factor model), or statistical.

Macroeconomic factor model

In this model, the factors ${\mathbf{f}{t}}$ are observed economic/financial time series out of given $\mathbf{X}$. The macroeconomic factor model can be estimated through Least-Squares (LS) regression: $$\underset{\boldsymbol{\gamma}{i}}{\mathsf{minimize}}\quad\Vert\mathbf{x}{i}-\tilde{\mathbf{F}}\boldsymbol{\gamma}{i}\Vert^{2}$$ where $\tilde{\mathbf{F}}=\left[\begin{array}{cc} \mathbf{1}{T} & \mathbf{F}\end{array}\right]$ and $\boldsymbol{\gamma}{i}=\left[\begin{array}{c} \alpha_{i}\ \boldsymbol{\beta}{i} \end{array}\right]$. The closed-form solution is: $\hat{\boldsymbol{\gamma}}{i}=\left(\tilde{\mathbf{F}}^{T}\tilde{\mathbf{F}}\right)^{-1}\tilde{\mathbf{F}}^{T}\mathbf{x}{i}$. Then simply use fator model to get $[\hat{\epsilon}{i,1},\dots,\hat{\epsilon}{i,T}]^T=\mathbf{x}{i}-\tilde{\mathbf{F}}\hat{\boldsymbol{\gamma}}_{i}$.

BARRA Industry factor model

Normally, fundamental factor model use observable asset specific characteristics (fundamentals) like industry classification, market capitalization, style classification (value, growth), etc. to determine the common risk factors $\mathbf{F}$. In this function, we only consider one of the cases: BARRA Industry factor model, which assumes that there are $K$ factors corresponding to $K$ mutually exclusive industries (aka, sectors). Apart from that, the loadings $\beta_{i,k}$ is directly defined as: $$\beta_{i,k}=\begin{cases} 1 & \textrm{if stock } i \textrm{ is in industry } k\ 0 & \textrm{otherwise.} \end{cases}$$ Using compact combination $\mathbf{B}=\left[\begin{array}{ccc} \boldsymbol{\beta}{1} & \cdots & \boldsymbol{\beta}{N}\end{array}\right]^{T}$, the industry factor model is (note that $\boldsymbol{\alpha} = \mathbf{0}$): $$\mathbf{x}{t} = \mathbf{B} \mathbf{f}{t} + \boldsymbol{\epsilon}{t}, \; t=1,\dots,T$$ where $\mathbf{x}{t} = [x_{1,t},\dots,x_{N,t}]^T$ and $\mathbf{f}{t} = [f{1,t},\dots,f_{K,t}]^T$. Here the LS regression can also be applied as: $$\underset{\mathbf{f}{t}}{\mathsf{minimize}}\quad\frac{1}{T}\sum{t=1}^{T}\Vert\mathbf{x}{t}-\mathbf{B}\mathbf{f}{t}\Vert_{2}^{2}$$ The solution is $\hat{\mathbf{f}}{t}=(\mathbf{B}^{T}\mathbf{B})^{-1}\mathbf{B}^{T}\mathbf{x}{t}, \; t=1,\dots,T$ and the residual can be simply calculated as $[\hat{\epsilon}{1,t},\dots,\hat{\epsilon}{N,t}]^{T}=\mathbf{x}{t}-\mathbf{B}\hat{\mathbf{f}}{t}$.

Statistical factor model

The statistical factor model holds the assumption that $\mathbf{f}{t}$ is affine transformations of $\mathbf{x}{t}$, i.e., $\mathbf{f}{t}=\mathbf{d}+\mathbf{C}^{T}\mathbf{x}{t}$, where $\mathbf{d}\in\mathbb{R}^{K}$ and $\mathbf{C}\in\mathbb{R}^{N\times K}$ are parameters to be estimated. We use iterative method [see slides_factor_model for details] to estimate parameters:

Calculate sample covariance matrix $\hat{\boldsymbol{\Sigma}}$ and its eigen-decomposition $\hat{\boldsymbol{\Gamma}}{1} \hat{\boldsymbol{\Lambda}}{1} \hat{\boldsymbol{\Gamma}}^{T}_{1}$, set index $s=1$

Estimate $\hat{\mathbf{B}}{(s)} = \hat{\boldsymbol{\Gamma}}{(s)} \hat{\boldsymbol{\Lambda}}^{\frac{1}{2}}{(s)}$, $\hat{\boldsymbol{\Psi}}{(s)} = \textrm{struct}(\hat{\boldsymbol{\Sigma}} - \hat{\mathbf{B}}{(s)} \hat{\mathbf{B}}^{T}{(s)})$, and $\hat{\boldsymbol{\Sigma}}{(s)} = \hat{\mathbf{B}}{(s)} \hat{\mathbf{B}}^{T}{(s)} + \hat{\boldsymbol{\Psi}}{(s)}$

Update the eigen-decomposition as $\hat{\boldsymbol{\Sigma}} - \hat{\boldsymbol{\Psi}}{(s)} = \hat{\boldsymbol{\Gamma}}{(s+1)} \hat{\boldsymbol{\Lambda}}{(s+1)} \hat{\boldsymbol{\Gamma}}^{T}{(s+1)}$ and $s \gets s+1$

Repeat Steps 2-3 until $(\hat{\mathbf{B}}{(s)}, \hat{\boldsymbol{\Psi}}{(s)}, \hat{\boldsymbol{\Sigma}}_{(s)})$ until convergence.

where $\textrm{struct}()$ is to impose certain structure on $\hat{\boldsymbol{\Psi}}{(s)}$, one typical option is diagonal. After the algorithm is done, we can calculate $\hat{\mathbf{\alpha}} = \frac{1}{T} \sum{t=1}^{T} \mathbf{x}{t}$ and build statistical factor model use algorithm output: $$ \hat{\mathbf{B}} = \hat{\boldsymbol{\Gamma}} \hat{\boldsymbol{\Lambda}}^{\frac{1}{2}}, \quad \hat{\mathbf{f}}{t} = \hat{\boldsymbol{\Lambda}}^{-\frac{1}{2}} \hat{\boldsymbol{\Gamma}}^{T} (\mathbf{x}{t} - \hat{\mathbf{\alpha}}), \quad \hat{\mathbf{\epsilon}}{t} = \mathbf{x}{t} - \hat{\mathbf{\alpha}} - \hat{\mathbf{B}} \hat{\mathbf{f}}{t}$$

`covFactorModel()`: Covariance matrix estimation through factor model

The function covFactorModel() estimate a covariance matrix based on factor model decomposition. As mentioned above, the factor model can be expressed as: $$\mathbf{x}{t} = \boldsymbol{\alpha} + \mathbf{B} \mathbf{f}{t} + \boldsymbol{\epsilon}{t}, \; t = 1, \dots, T$$ Usually, we will assume ${\mathbf{f}{t}}$ and ${\boldsymbol{\epsilon}{t}}$ are uncorrelated. Then the covariance matrix $\boldsymbol{\Sigma}$ can be represented as: $$\boldsymbol{\Sigma} = \mathbf{B} \boldsymbol{\Sigma}{\mathbf{f}} \mathbf{B}^{T} + \boldsymbol{\Psi}$$ where $\boldsymbol{\Sigma}{\mathbf{f}} = \mathsf{Cov}[\mathbf{x}{t}]$ and $\boldsymbol{\Psi} = \mathsf{Cov}[\boldsymbol{\epsilon}{t}]$. We can simply use result from function factorModel() to estimate covariance matrix $\boldsymbol{\Sigma}$ as: $$\hat{\boldsymbol{\Sigma}} = \hat{\mathbf{B}} \hat{\boldsymbol{\Sigma}}{\mathbf{f}} \hat{\mathbf{B}}^{T} + \hat{\boldsymbol{\Psi}}$$ where $\hat{\boldsymbol{\Sigma}}{\mathbf{f}}$ and $\hat{\boldsymbol{\Psi}}$ are the sample covariance matrix of ${\mathbf{\mathbf{f}}{t}}$ and ${\boldsymbol{\epsilon}{t}}$. Besides, the $\boldsymbol{\Psi}$ is expected to follow a special structure, i.e., $$\hat{\boldsymbol{\Sigma}} = \hat{\mathbf{B}} \hat{\boldsymbol{\Sigma}}{\mathbf{f}} \hat{\mathbf{B}}^{T} + \textrm{struct}{ \hat{\boldsymbol{\Psi}} }$$ In statistical factor model, the $\hat{\boldsymbol{\Sigma}}$ is actually available when build that model. Therefore the algorithm output $\hat{\boldsymbol{\Sigma}}_{(s)}$ will be directly extracted as covariance matrix estimation.