Factor Models

• Goal: How can we explain the data using components that maximize the signal while minimizing the noise?
  • Signal: unique information stored in each variables, the variance of certain variables.
  • Noise: redundant (overlapping) information in a certain variables, the covariance of the variables with others.
• Usage: To extract the co-factors that can explain the group of variables.
  • (ex; Output) multiple indices, but shared characteristics?
  • (ex; Price) hundreds of items, but driven by a small number of factors?
• Limitation: Can distort the original information.
  • How should we interpret the extracted components?
  • Might not provide a stronger explanatory power than just a weighted sum or arbitrarily selected variables.

Let the population data is given as then the population covariance matrix is which positive definite.

Our goal is to find that are the linear combinations of variables, such that

1. maximizes , where
2. each () are uncorrelated, i.e.
3. coefficients are normalized to solve the problem, i.e.

We need to solve First, we calculate : then the Lagrangian function is F.O.C. thus is a unit eigenvector of that corresponds to the largest eigenvalue .

Next, we calculate : then the Lagrangian function is F.O.C. note that since , we have , as we are assuming that . therefore we have thus we have , i.e. is a unit eigenvector of that corresponds to that is orthogonal to .

Similarly, we can derive PCs, such that corresponds to .

• Number of PCs is : where are the orthonormal eigenbasis.
• Let , then we have
• Strength of -th PC: .

Let be the given sample data: i.e. for each variable , we have samples.

1. Calculate the sample covariance, which will converge in probability to by Law of Large Numbers.
2. Choose the orthogonal eigenbasis of with ordered eigenvalues , so that
3. Since are sample variances of each sample PC, we usually normalize by
4. Finally, as the orthogonal eigenvectors of the positive definite matrix are not unique, apply additional restrictions to identify PCs.

• Goal: find the small number of factors that can explain the co-movement of the economic variables.
• Static Factor Model: dynamics of the factors ignored.
• exact factor model: completely explains cross-sectional and temporal co-movement, while is purely idiosyncratic.
• approximate factor model: limited correlation among is allowed.

Let the data be matrix: or equivalently, note that .

LSAFM follows in matrix form: or equivalently,

In vector form: or equivalently,

Then PC estimator of minimizes the objective function : The PC estimator can be driven in either two approaches:

1. First concentrate out the factor loadings, and then obtain the factors.
2. First concentrate out the factors, and then obtain the loadings.

For the convenience of the matrix computations, the first approach is preferred when , and the later one is often used when . We first look for the second method, but the first approach follows the similar proceedings to the second one. Furthermore, the PCs obtained from either approach are closely related, which are shown in Bai and Ng (2002).

• (MSE) find such that minimize
• (ER) find that maximize .
• (PIC) find that minimize , where commonly we use .