18-MultiFactorModels Page 543 Wednesday, February 4, 2004 1:10 PM
Multifactor Models and Common Trends for Common Stocks 543The static-factor model and the common-trend cointegrated model are
special cases of the above general dynamic factor model. The ARDL
model is a dynamic factor model with additional restrictions.
A conceptual parallel can be made between state-space models and fac-
tor models. Recall that factor models essentially address the problem of a
nearly random cross-correlation matrix. The correlation coefficients of a
large correlation matrix are essentially random. To recover a meaningful
correlation structure, every process is represented as a linear regression on
a set of factors and only correlations between factors are considered.
Were we to attempt an estimate of a global VAR model of a large
portfolio of equity prices or return processes, we would run into the same
problem of finding meaningless random autocross correlation coefficients.
This is because the matrices that represent all the correlations at different
time lags are nearly random. State-space models extract the useful auto-
correlation information from a large set of auto-cross-correlation data.Estimation and Testing of Cointegrated Systems
The estimation and testing of cointegrated systems is a complex issue on
which there is vast literature. The two major methods for estimation of
cointegrated systems are due to Engle and Granger^13 and Johansen.^14 The
Engle-Granger method is based on writing down explicitly the long-run
regression equation and subsequently estimating the short term correc-
tions. The Johansen methodology applies directly MLE methods. Stock
and Watson^15 proposed PCA to determine the common trends.^16
When dealing with large sets of asset prices, in particular equity
prices, the techniques of Engle-Granger and Johansen are not applica-
ble. The PCA-based approach of Stock and Watson, on the other hand,
can be applied to hundreds of price processes. The Stock and Watson
methodology is based on the observation that if there are r cointegra-
tion relationships the resulting n-r common trends are integrated I(1)
while the r cointegrating portfolios are stationary I(0). Consequently, it
is reasonable to assume that the integrated portfolios have maximum
variance. Therefore, performing PCA on the variance-covariance matrix
of the price process should lead to identification of the number and the(^13) Engle and Granger, “Cointegration and Error Correction: Representations, Esti -
mation and Testing.”
(^14) S. Johansen, Likelihood-based Inference in Cointegrated Vector Autoregressive
Models (Oxford: Oxford University Press, 1995).
(^15) Stock and Watson, “Testing for Common Trends.”
(^16) The interested reader should consult the original works quoted or, G.S. Maddala
and In-Moo Kim, Unit Roots, Cointegration, and Structural Changes (Cambridge,
U.K.: Cambridge University Press, 1988).