Tutorial lecture 3

1Tutorial lecture 3Reducing the dimension of the parameter space: FactorModelsModeling of comovement or of relations between single time series inmultivariate time series. Here we consider (static and dynamic)• Principal component models• Frisch or idiosyncratic noise model• Reduced rank regressionECONOMETRIC FORECASTING AND HIGH FREQUENCY DATA ANALYSIS, Singapore, May 200423.1 The basic framework:We restrict ourselves to the stationary case:0y = Λ(z)ξ +u , Eξ u = 0 (1)t t t t swherey ... observations (n–dim.)tξ ... factors (unobserved) (r < 0,f (λ)> 0, rkΛ =ry ξfor the quasi static case we obtain∗ 0Σ = ΛΣ Λ +Σ where e.g. Σ =Ey y (3)y ξ u y t tIdentifiability questions:∗• Identifiability off = Λf Λ andfyˆ ξ u• of Λ andfξECONOMETRIC FORECASTING AND HIGH FREQUENCY DATA ANALYSIS, Singapore, May 200464Estimation of integers and real valued parameters:• Estimation of r• Estimation of the free parameters in Λ,f ,fξ u• Estimation ofξtForecasting model for factors:0ξ =a(z)ξ +d(z)x + , ( ) white noise,Ex = 0 (4)t+1 t t t+1 t t sstability condition: det(I−za(z)) = 0 |z|≤ 1ECONOMETRIC FORECASTING AND HIGH FREQUENCY DATA ANALYSIS, Singapore, May 200453.2 Principal Component Analysis1. ...