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Tutorial lecture 3

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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. ...
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Tutorial lecture 3 Reducing the dimension of the parameter space: Factor Models
Modeling of comovement or of relations between single time series in multivariate time series. Here we consider (static and dynamic)
Principal component models
Frisch or idiosyncratic noise model
Reduced rank regression
1
ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
3.1 The basic framework: We restrict ourselves to the stationary case:
y t = Λ( z ) ξ t + u t E ξ t u 0 s = 0
2
(1)
where y t    observations (n–dim.) ξ t    factors (unobserved) ( r << n –dim.) Λ( z ) = P j = −∞ Λ j z j , Λ j R n × r    factor loadings y ˆ t = Λ( z ) ξ t    latent variables Λ = Λ 0    (quasi-static) case.
ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
Spectral densities:
f y = Λ f ξ Λ + f u
Ass.: f y ( λ ) > 0 , f ξ ( λ ) > 0 , rk Λ = r
for the quasi-static case we obtain
Σ y = ΛΣ ξ Λ + Σ u where e.g. Σ y = E y t y t 0
Identiability questions:
Identiability of f y ˆ = Λ f ξ Λ and f u
Identiability of Λ and f ξ
3
(2)
(3)
ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
Estimation of integers and real-valued parameters:
Estimation of r
Estimation of the free parameters in Λ , f ξ , f u
Estimation of ξ t
Forecasting model for factors:
ξ t +1 = a ( z ) ξ t + d ( z ) x t + t +1 ( t ) white noise, E x t 0 s = 0
stability condition: det( I za ( z )) 6 = 0 | z | ≤ 1
4
(4)
ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
3.2 Principal Component Analysis 1. The quasi-static case: Eigenvalue decomposition of Σ y :
Σ y = O Ω O 0 = | O 1 Ω z 1 O 0 1 } + | O 2 Ω {z 2 O 0 2 } { Σ y ˆ Σ u where Ω 1 is the r × r –dim. diogonal matrix containing the r largest eigenvalues of Σ y . This decomposition is unique for ω r > ω r +1 . A special choice for the factor loading matrix is Λ = O 1 , then
y t = O 1 ξ t + u t
5
ξ t = O 0 1 y t , u t = y t O 1 O 0 1 y t = O 2 O 0 2 y t Note: Factors are linear functions of y t . ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
Estimation:
Determine r from ω 1      ω n Estimate Λ Σ ξ Σ u  ξ t from the eigenvalue decomposition of Σˆ y = T 1 P tT =1 y t y 0 t
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ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
2. The dynamic case: We commence from the spectral density f y rather than from Σ y
f y ( λ ) = | O 1 ( λ 1 { ( z λ ) O 1 ( λ } ) + | O 2 ( λ 2 { ( z λ ) O 2 ( λ } ) f y ˆ ( λ ) f u ( λ )
y t = O 1 ( z ) ξ t + u t
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then ξ t = O 1 ( z ) y t Note: Here E u 0 t u t is minimal among all decompositions where rk (Ω( z )) = r a.e. Again ξ t = O 1 ( z ) y t , i.e. factors are linear transformations of ( y t ) Problem: In general, the lter O 1 ( z ) will be non-causal and non-rational. Thus, naive forecasting may lead to infeasible forecasts for y t . Restriction to causal lters is required. In estimation, we commence from a spectral estimate. ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
3.3 The Frisch model
Here the additional assumption f u is diagonal is imposed in ( 1 ).
Interpretation: Factors describe the common effects, the noise u t takes into account the individual effects, e.g. factors describe markets and sector specic movements and the noise the rm specic movements of stock returns.
For given y ˆ t the components of y t are conditionally uncorrelated.
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ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
1. The quasi-static case:
Identiability:More demanding compared to PCA
Identiability of Σ y ˆ :
Σ y = | ΛΣ {z ξ Λ } 0 u u ) digaonal Σ y ˆ
Uniqueness of solution of ( 5 ) for given n and r, the number of equations (i.e. the number of free elements in Σ y ) is n ( n 2+1) . The number of free parameters on the r.h.s. is nr r ( r 2 1) + n . Now let
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(5)
n ( n + 1) B ( r )2 ( nr r ( r 2 1)+ n )=12(( n r ) 2 n r ) = ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
then the following cases may occur:
B ( r ) < 0 : In this case we might expect non-uniqueness of the decomposition B ( r ) 0 : In this case we might expect uniqueness of the decomposition
The argument can be made more precise, in particular, for B ( r ) > 0 generic uniqueness can be shown.
Given Σ y , if Σ ξ = I r is assumed, then Λ is unique up to postmultiplication by orthogonal matrices (rotation).
10
Note that, as opposed to PCA, here the factors ξ t , in general, cannot be obtained as a function of the observations y t . Thus, the factors have to be approximated by a linear function of y t . The following two approximations are used:
ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
1. The regression method investigated by Thomson: 11 The idea, here, is to estimate ξ t by a linear function of y t such that the ˆ ˆ variance of the estimation error, ξ t ξ t , is minimal. Therefore, ξ t is given by the regression of ξ t onto y t ,
ξ ˆ tT = Λ 0 Σ y 1 y t
since by the above assumptions
E y t ξ t 0 = E [(Λ ξ t + u t ) ξ 0 t ] = Λ
As can easily be seen, this estimator is biased in a certain sense, since E ( ξ ˆ tT | ξ t ) = Λ 0 Σ y 1 ξ t + E ( u t | ξ t )) 6 = ξ t .
(6)
(7)
ECONOMETRIC FORECASTING AND HIGH-FREQUENCY DATA ANALYSIS, Singapore, May 2004
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