🗊Презентация Basics of factor models

Why factor models?

Why factor models?

What can be done with factor models?

An introduction to factor models

Some extensions

Representation

Representation

Identification

Factor models and VARs

Estimation by the Kalman filter

Non-parametric, large N, factor models

The SW approach - PCA The Stock and Watson (2002a,2002b) factor model is Xt = Λft ‡ ξt , where: Xt is N × 1 vector of stationary variables ft is r × 1 vector of common factors, can be correlated over time Λ is N × r matrix of loadings ξt is N × 1 vector of idiosyncratic disturbances, can be mildly cross-sectionally and temporally correlated conditions on Λ and ξt guarantee that the factors are pervasive (affect most variables) while idiosyncratic errors are not.

The SW approach - PCA

The SW approach - Choice of r

The SW approach - Properties of PCA

The SW approach - Properties of PCA

The SW approach - Properties of PCA based forecasts

The FHLR approach - DPCA

The FHLR approach - static and dynamic factors

The FHLR approach - Choice of q

The FHLR approach - Forecasting

Parametric estimation - quasi MLE

Parametric estimation - quasi MLE

Parametric estimation - Subspace algorithms (SSS)

Parametric estimation - SSS

Parametric estimation - SSS forecasts

Factor estimation methods - Monte Carlo Comparison

Factor estimation methods - MC Comparison, summary

Factor models - Forecasting performance Really many papers on forecasting with factor models in the past l5 years, starting with Stock and Watson (2002b) for the USA and Marcellino, Stock and Watson (2003) for the euro area. Banerjee, Marcellino and Masten (2006) provide results for ten Eastern European countries. Eickmeier and Ziegler (2008) provide nice summary (meta-analysis), see also Stock and Watson (2006) for a survey of the earlier results. Recently used also for nowcasting, i.e., predicting current economic conditions (before official data is released). More on this in the next lecture.

Factor models - Forecasting performance

Structural Factor Augmented VAR (FAVAR)

Structural FAVAR

Structural FAVAR - Monetary policy shock identification

Structural FAVAR - Monetary policy shock identification

Structural FAVAR - Monetary policy shock identification

Structural FAVAR - Monetary policy (FFR) shock

Structural FAVAR - Monetary policy (FFR) shock

Structural FAVAR - Monetary policy (FFR) shock

Structural FAVAR: Summary

References

Stock, J.H. and Watson, M. W. (20l5), “Factor Models for Macroeconomics," in J. B. Taylor and H. Uhlig (eds), Handbook of Macroeconomics, Vol. 2, North Holland.

The FHLR approach - DPCA

The FHLR approach - DPCA

Parametric estimation - Subspace algorithms (SSS)

Parametric estimation - SSS, T asymptotics

Parametric estimation - SSS, T and N asymptotics

Factor estimation methods - MC Comparison First set of experiments: a single VARMA factor with di"erent specifications: 1a1 =0.2, b1 = 0.4¡ 2 a1 =0.7, bl =0.2¡ 3 a1 =0.3, a2 = 0.1, b1 = 0.15, b2 = 0.15¡ 4 a1 = 0.5, a2 = 0.3, b1 = 0.2, b2 = 0.2¡ 5 a1 = 0.2, b1 = —0.4¡ 6 a1 = 0.7, b1 = —0.2¡ 7 a1 = 0.3, a2 = 0.1, b1 = —0.15, b2 = —0.15¡ 8 a1 = 0.5, a2 = 0.3, b1 = —0.2, b2 = —0.2. 9 As 1but C = C0 + C1L. 10 As 1but one factor assumed instead of p + q

Factor estimation methods - MC Comparison Second group of experiments: as in 1-10 but with each idiosyncratic error being an AR(1) process with coefficient 0.2 (exp. 11-20). Experiments with cross correlation yield similar ranking of methods. Third group of experiments: 3 dimensional VAR(1) for the factors with diagonal matrix with elements equal to 0.5 (exp. 21). Fourth group of experiments: as 1-21 but the C matrix is U(0,1) rather than N(0,1). Fifth group of experiments: as 1-21 but using s = 1instead of s = m.

Factor estimation methods - MC Comparison

Factor estimation methods - MC Comparison, N=T=50 Single ARMA factor (exp. 1-8): looking at correlations, SSS clearly outperforms PCA and DPCA. Gains wrt PCA rather limited, 5-10%, but systematic. Larger gains wrt DPCA, about 20%. Little evidence of correlation of idiosyncratic component , but rejection probabilities of LM(4) test systematically larger for DPCA. Serially correlated idiosyncratic errors (exp. 11-18): no major changes. Low rejection rate of LM(4) test due to low power for T = 50. Dynamic effect of factor (exp. 9 and l9): serious deterioration of SSS, a drop of about 25% in the correlation values. DPCA improves but it is still beaten by PCA. Choice of s matters: for s =1SSS becomes comparable with PCA (Table 9).

Factor estimation methods - MC Comparison, N=T=50

Factor estimation methods - MC Comparison, other results