10.4.3 Missing Observations

10.4.3 Missing Observations

Incomplete data and missing observations are important practical concerns for economic forecasters. Far from being a balanced panel of $T\times N$ observations, the typical data set encountered in practice involves variables whose samples are of different length and may even be measured at different frequencies. To address this problem, Stock and Watson (2002b) propose using a simple expectation-maximization (EM) algorithm that fills in missing observations. Consider the objective function

$$V(F^P,\Lambda )=\sum _{i=1}^N\sum _{t=1}^T{1}_{it}(X_{it}-{\lambda }_i^{\prime }{F}_t{)}^2,\text{\hspace{1em}(10.27)}$$

where ${\lambda }_i$ is the i th row of - and $1_{it}$ is an indicator function defined as follows:

$$1_{it}=\{\begin{array}{ll}1& \mathrm{if}{x}_{it}\mathrm{is}\mathrm{observed},\\ 0& \mathrm{if}{x}_{it}\mathrm{is}missing.\end{array}$$

Stock and Watson propose an iterative procedure for obtaining an initial set of estimates, ${\hat{\lambda }}_i$ , and the accompanying factors, ${\hat{F}}_t^{}.$ . These are then used to fill out the missing values, i.e., ${\hat{X}}_{it}=1_{it}{X}_{it}+(1-1_{it}){\hat{\lambda }}_i^{\prime }{\hat{F}}_t$ . Using the rebalanced data set, a new set of factor estimates, ${\hat{F}}_t,$ is extracted from the largest eigenvalues of the sample covariance matrix, $N^{-1}{\sum }_{i=1}^N{\hat{X}}_i{\hat{X}}_i^{\prime }$ , where ${\hat{X}}_i=({\hat{X}}_{i1},\dots ,{\hat{\hat{X}}}_{iT}{)}^{\prime }$ . These factor estimates are in turn used to obtain new estimates of ${\hat{\lambda }}_i$ by regressing $\hat{X}$ on the updated factor estimates and updating the missing observations. The process continues until convergence. For further details, see Stock and Watson (2002b, Appendix A).

In addition to missing data points, outliers can be a real problem when dealing with large data sets for which every series cannot be guaranteed to be of the same quality as when only a few series are used. A filter that replaces outliers with an estimate can be adopted in such situations.

10.4.4 Partial Least Squares

Principal components methods ignore information on the outcome variable, y, in constructing linear combinations of the N predictors, X. The partial least squares (PLS) method accounts for such correlations in constructing linear combinations of the predictors. The method first standardizes each x-variable to have zero mean and unit variance, i.e., ${\tilde{x}}_i=(x_i-{\bar{x}}_i)/{\hat{\sigma }}_i.$ where ${\bar{x}}_i,{\hat{\sigma }}_i$ are the sample mean and sample standard deviation of $x_i.$ , respectively. The method then proceeds iteratively through the following algorithm as described by Hastie, Tibshirani, and Friedman (2009, page 81). First, initiate the algorithm by setting $f^0=\bar{y}{\iota }_T$ , where $\bar{y}=T^{-1}{\sum }_{t=1}^T{y}_t$ and ${\tilde{x}}_i^0={\tilde{x}}_i$ . Then, for $m=1,\dots ,\bar{m}\le N$ repeat the following steps: (i) form the linear combination of x-variables $\begin{array}{r}{z}_m={\sum }_{j=1}^N{\bar{\hat{\beta }}}_{jm}{\tilde{x}}_j^{m-1}\end{array}$ , where ${\hat{\beta }}_{jm}=\mathrm{Cov}({\tilde{x}}_j^{m-1},y)$ ; (ii) update the forecast to $f^m=f^{m-1}+{\hat{\theta }}_m{z}_m$ , where ${\hat{\theta }}_m=\widehat{\mathrm{Cov}}(z^m,y)/\widehat{\mathrm{Var}}(z^m);(\mathrm{iii})$ orthogonalize ${\tilde{x}}_j^{m-1}$ with respect to $z_m$ by setting ${\tilde{x}}_j^m={\tilde{x}}_j^{m-1}-{\hat{\gamma }}_j^m{z}_m$ , where ${\hat{\gamma }}_j^m=$ $\widehat{\mathrm{Cov}}(z_m,x_i^{m-1})/\widehat{\mathrm{Var}}(z_m)$ . Then repeat step (i).

Here $z_m$ are linear combinations of the x˜-variables and are often referred to as directions. Notice how the partial least squares method weights the individual $\tilde{x}-$ variables by the strength of their (univariate) correlations with the outcome. If the algorithm continues until $m=N$ , it becomes similar to OLS regression on the full set of original predictors.

Despite explicitly accounting for the correlation between the outcome and the linear combinations of x-variables, in practice, often the PLS method acts similarly to Ridge regressions and principal components methods; for further discussion, see Hastie, Tibshirani, and Friedman (2009).