What is the difference between PCA and asymptotic PCA?

Question

In two papers in 1986 and 1988, Connor and Korajczyk proposed an approach to modeling asset returns. Since these time series have usually more assets than time period observations, they proposed to perform a PCA on cross-sectional covariances of asset returns. They call this method Asymptotic Principal Component Analysis (APCA, which is rather confusing, since the audience thinks immediately of asymptotic properties of PCA).

I have worked out the equations, and the two approaches seem numerically equivalent. The asymptotics of course differ, since convergence is proved for $N \rightarrow \infty$ rather than $T \rightarrow \infty$. My question is: has anyone used APCA and compared to PCA? Are there concrete differences? If so, which ones?

Answer 1

There is absolutely no difference.

There is absolutely no difference between standard PCA and what C&K suggested and called "asymptotic PCA". It is quite ridiculous to give it a separate name.

Here is a short explanation of PCA. If centered data with samples in rows are stored in a data matrix $\mathbf X$, then PCA looks for eigenvectors of the covariance matrix $\frac{1}{N}\mathbf X^\top \mathbf X$, and projects the data on these eigenvectors to obtain principal components. Equivalently, one can consider a Gram matrix, $\frac{1}{N}\mathbf X \mathbf X^\top$. It is easy to see that is has exactly the same eigenvalues, and its eigenvectors are scaled PCs. (This is convenient when the number of samples is less than the number of features.)

It seems to me that what C&K suggested, is to compute eigenvectors of the Gram matrix in order to compute principal components. Well, wow. This is not "equivalent" to PCA; it is PCA.

To add to the confusion, the name "asymptotic PCA" seems to refer to its relation to factor analysis (FA), not to PCA! The original C&K papers are under paywall, so here is a quote from Tsay, Analysis of Financial Time Series, available on Google Books:

Connor and Korajczyk (1988) showed that as $k$ [number of features] $\to \infty$ eigenvalue-eigenvector analysis of [the Gram matrix] is equivalent to the traditional statistical factor analysis.

What this really means is that when $k \to \infty$, PCA gives the same solution as FA. This is an easy-to-understand fact about PCA and FA, and it has nothing to do with whatever C&K suggested. I discussed it in the following threads:

• Is there any good reason to use PCA instead of EFA? Also, can PCA be a substitute for factor analysis?
• Under which conditions do PCA and FA yield similar results?

So the bottom-line is: C&K decided to coin the term "asymptotic PCA" for standard PCA (which could also be called "asymptotic FA"). I would go as far as to recommend never to use this term.

Answer 2

Typically APCA gets used when there are lots of series but very few samples. I wouldn't describe APCA as better or worse than PCA, because of the equivalence you noted. They do, however, differ in when the tools are applicable. That is the insight of the paper: you can flip the dimension if it's more convenient! So in the application you mentioned, there are a lot of assets so you would need a long time series to compute a covariance matrix, but now you can use APCA. That said, I don't think APCA gets applied very often because you could try to reduce the dimensionality using other techniques (like factor analysis).

Answer 3

It's not about Math. only. To understand why it's different, you need to know the Finance theory background of factor models. For a factor model $R = BF + \epsilon$. It is like a multivariate regression, R has dimension the number of stocks.

• F, the number of factors.
• B is a matrix of loadings.
• $\epsilon$ (epsilon) has a diagonal covariance matrix.

In PCA you first get the loadings as the Eigen vectors of the covariance matrix. Then you decide how many you keep, say K. In a second step you get the K factor scores (estimates of the K factor values at every time t) by cross-sectional regression of the stock returns on the loadings for every time (aka Bartlett method). Since the Bs are noisy estimates from a noisy covariance matrix, you have a measurement error in your right hand side data (the Bs) in the second step.

Now why it's important: The typical equity finance application has many many stocks and not so many time period => noisy noisy cov matrix => noisy loadings B (can you say Junk in?) => massive bias to zero of the estimates of the factor scores. In the extreme, you have more stocks than periods and your cov. matrix is not full rank.

In Asymptotic PC, see Connors and Korajczik, the cross-product matrix is TxT, where T is the number of observations. It is always PDS, as the number of stocks goes to infinity for a given sample size, it converges to the factor scores with arbitrary precision. Then in a second step you get the loadings by a time series regression of each return on the factors.

And here is the (nice) catch: In a stock situation, you have N >> T, you could not even estimate the cov.matrix properly. But you have infinitely precise estimates of the factor scores. In the second step where you get the loadings from the scores, you have NO problem of errors in the variables.

If you take the pain to read their papers, you will see the simulation that shows how effective this is for typical cross-section size vs sample size.