# RDP 2007-03: Forecasting with Factors: The Accuracy of Timeliness Appendix B: Alternative Factor Estimates

An alternative technique for estimating the factors has been developed by Forni et al (2005) (FHLR). Their methodology takes into account the possibility of leading and lagging relationships of the series in the data panel, and so is referred to as being ‘dynamic’ (while the technique we use in the paper is referred to as ‘static’). The estimation of the dynamic model is more complex than that of the static model, as the factors are estimated in the frequency domain rather than the time domain. While the steps involved in the two estimation procedures differ, conceptually they are still closely related. As Stock and Watson (2006) note, while principal components of the static approach has a least squares interpretation, the dynamic approach has a weighted least squares interpretation. Boivin and Ng (2005) succinctly describe the steps involved in estimating the dynamic factors and provide a comparison of the static and dynamic methods. We use their notation in this section.

We present two different ways of forecasting with the dynamic factors. The first, denoted FHLR-DU, estimates the dynamic factors and then uses them in the forecasting equation, Equation (2), as done with the static factors in Section 2.2. The second, non-parametric, approach produces a forecast directly by projecting forward the common component for each forecast series, and is denoted FHLR-DN. Which technique produces more accurate forecasts is ultimately an empirical issue. Table B1 reproduces the results from Table 2 in Section 4 along with equivalent results for these two dynamic factor methods.

Model Horizon Two quarters Four quarters Eight quarters GDP growth SW FHLR-DU F2 0.92 (0.12) 0.85 (0.14) 0.83* (0.13) FAR2 0.96 (0.17) 0.89 (0.17) 0.84 (0.14) FAR-BIC 1.10 (0.14) 1.02 (0.16) 0.78* (0.14) F2 0.99 (0.11) 0.91 (0.13) 0.79* (0.14) FAR2 1.00 (0.16) 0.94 (0.16) 0.79* (0.15) FAR-BIC 1.11 (0.17) 1.06 (0.19) 0.75* (0.16) 0.88** (0.07) 0.86* (0.09) 0.74** (0.14) F2 0.78** (0.12) 0.76* (0.15) 0.73* (0.17) FAR2 0.86 (0.15) 0.87 (0.14) 0.74* (0.17) FAR-BIC 0.95 (0.14) 0.93 (0.14) 0.78* (0.15) F2 0.86* (0.10) 0.81* (0.13) 0.68** (0.18) FAR2 0.91 (0.14) 0.92 (0.12) 0.68** (0.18) FAR-BIC 0.84 (0.13) 0.83 (0.15) 0.70** (0.18) 0.80** (0.09) 0.80** (0.11) 0.71** (0.16) F2 0.82* (0.12) 0.71** (0.15) 0.77* (0.15) FAR2 0.78** (0.13) 0.67** (0.17) 0.73* (0.17) FAR-BIC 0.81* (0.13) 0.74* (0.18) 0.80 (0.16) F2 0.86 (0.11) 0.77** (0.12) 0.73** (0.14) FAR2 0.80** (0.12) 0.71** (0.15) 0.74** (0.15) FAR-BIC 0.83* (0.12) 0.63** (0.19) 0.72* (0.18) 0.79** (0.09) 0.81** (0.10) 0.78** (0.13) F2 0.90* (0.07) 0.69*** (0.11) 0.64** (0.16) FAR2 0.97 (0.08) 0.69*** (0.11) 0.65** (0.16) FAR-BIC 1.20 (0.26) 0.97 (0.21) 1.00 (0.26) F2 0.93 (0.07) 0.72*** (0.11) 0.64*** (0.15) FAR2 0.97 (0.07) 0.73*** (0.11) 0.66** (0.15) FAR-BIC 1.20 (0.29) 0.97 (0.22) 0.89 (0.16) 0.89** (0.05) 0.79*** (0.08) 0.67** (0.14) F2 0.76** (0.14) 0.71** (0.16) 0.77* (0.17) FAR2 0.78** (0.13) 0.76* (0.17) 0.80 (0.18) FAR-BIC 0.82* (0.12) 0.84 (0.14) 0.81 (0.17) F2 0.78** (0.13) 0.73** (0.15) 0.75* (0.17) FAR2 0.78** (0.13) 0.77* (0.15) 0.81 (0.18) FAR-BIC 0.85 (0.12) 0.81 (0.15) 0.77 (0.18) 0.83** (0.10) 0.80* (0.11) 0.79** (0.12) F2 16.02 (4.06) 4.08 (2.65) 1.80 (0.61) FAR2 0.67* (0.16) 0.71* (0.16) 0.76* (0.13) FAR-BIC 0.58* (0.19) 0.70* (0.17) 0.71* (0.15) F2 16.33 (37.92) 4.06 (2.88) 1.68 (0.55) FAR2 0.71* (0.15) 0.73* (0.15) 0.70* (0.14) FAR-BIC 0.59* (0.19) 0.70* (0.18) 0.68* (0.16) 6.14 (5.93) 2.22 (0.78) 1.39 (0.24) F2 1.78 (0.56) 1.63 (0.40) 1.14 (0.22) FAR2 0.89 (0.12) 0.84 (0.16) 0.71* (0.20) FAR-BIC 1.01 (0.20) 0.78 (0.21) 0.63* (0.28) F2 1.92 (0.69) 1.61 (0.36) 1.14 (0.19) FAR2 1.08 (0.16) 0.88 (0.19) 0.74 (0.23) FAR-BIC 1.02 (0.19) 0.86 (0.22) 0.78 (0.27) 2.22 (0.80) 1.57 (0.41) 1.20 (0.17) F2 1.09 (0.10) 1.04 (0.09) 0.94 (0.11) FAR2 1.01 (0.08) 1.04 (0.09) 0.98 (0.12) FAR-BIC 0.98 (0.09) 0.98 (0.09) 1.01 (0.11) F2 1.19 (0.11) 1.11 (0.10) 0.93 (0.10) FAR2 1.07 (0.08) 1.05 (0.09) 1.00 (0.11) FAR-BIC 0.89 (0.09) 0.90 (0.09) 0.98 (0.10) 1.05 (0.06) 1.03 (0.06) 1.10 (0.09) Notes: SW indicates factors estimated using the technique of Stock and Watson (1999; 2002a; 2002b). FHLR indicates factors estimated using the technique of Forni, Hallin, Lippi and Reichlin (2005). In the notation of Boivin and Ng (2005), DU indicates ‘dynamic unrestricted’ factors, while DN indicates ‘dynamic non-parametric factors’. Model F2 includes two factors and no lags of the forecast variable. Model FAR2 includes two factors and up to four lags of the forecast variable, selected at each iteration using the BIC (0 ≤ p ≤ 3). Model FAR-BIC uses the BIC to select both the number of factors (up to six, 1 ≤ k ≤ 6) and the number of lags of the forecast variable (up to three, 0 ≤ p ≤ 3) at each iteration. Numbers in parentheses are robust standard errors calculated using the delta method. Ratios significantly less than 1 at the 1, 5 and 10 per cent confidence levels are indicated by ***, ** and *.

The difference in forecast accuracy between the Stock and Watson static (SW) and FHLR-DU forecasting techniques is typically less than 5 per cent relative to the autoregressive benchmark for all forecasting model specifications. There is some evidence that the SW model performs better at the two- and four-quarter horizons, and the FHLR-DU model at the eight-quarter horizon for the series we forecast, but the difference in forecast accuracy is very small.

Aside from the unemployment rate and CPI series, there is also little difference in forecast accuracy between the SW and FHLR-DN techniques. For the unemployment rate and CPI series, the FHLR-DN techniques forecasts are substantially worse than those made using the SW technique, most likely because the FHLR-DN technique cannot capture the persistence in these series, which the SW and FHLR-DU forecasting models can with the incorporation of autoregressive terms in the forecasting equation. Indeed, the forecast accuracy of the SW technique is comparable to, if not worse than, the FHLR-DN technique for the unemployment rate and CPI series when autoregressive terms are not included in the forecasting equation.

These results confirm the findings of other authors that the simpler SW technique can be used to generate forecasts that are typically at least as accurate as those made using either of the more complicated FHLR techniques. Other forecasts, not reported here, show that the results in the text indicating forecast accuracy conditional on the timeliness of the data are qualitatively similar using the FHLR-DU technique. Because the FHLR-DN technique requires the forecast series to be in the data panel at all times, it is not suitable for our evaluation of forecast accuracy conditional on the timeliness of the data panel.