TYPES OF FACTOR MODELS
In this section we describe the different types of factor models.
Known Factors
The simplest case of factor models is where the K factors are assumed known or observable, so that we have time-series data on them. In this case, the K-factor model for the return-generating process as given by equation (5.2) is a multiple regression for each asset, and is a multivariate regression if all of the individual regressions are pooled together. For example, if one believes that the gross domestic product (GDP) is the driving force for a group of stock returns, one would have a one-factor model,
The above equation corresponds to equation (5.1) with K = 1 and 
In practice, one can obtain time-series data on both the asset returns and GDP, and then one can estimate the regressions to obtain all the parameters, including in particular the expected returns.
In practice, one can obtain time-series data on both the asset returns and GDP, and then one can estimate the regressions to obtain all the parameters, including in particular the expected returns.
Another popular one-factor model is the market model regression
where 
is the return on a stock market index.
is the return on a stock market index.
To understand the covariance matrix estimation, it will be useful to write the K-factor model in matrix ...
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