Factor Models: A Review James J. Heckman The University of Chicago - PowerPoint PPT Presentation

Factor Models: A Review James J. Heckman The University of Chicago Econ 312, Winter 2019 Heckman Factor Models: A Review of General Models

Factor Models: A Review of General Models Heckman Factor Models: A Review of General Models

E ( θ ) = 0; E ( ε i ) = 0; i = 1 , . . . , 5 R 1 = α 1 θ + ε 1 , R 2 = α 2 θ + ε 2 , R 3 = α 3 θ + ε 3 , R 4 = α 4 θ + ε 4 , R 5 = α 5 θ + ε 5 , ε i ⊥ ⊥ ε j , i � = j Cov ( R 1 , R 2 ) = α 1 α 2 σ 2 θ Cov ( R 1 , R 3 ) = α 1 α 3 σ 2 θ Cov ( R 2 , R 3 ) = α 2 α 3 σ 2 θ • Normalize α 1 = 1 Cov ( R 2 , R 3 ) Cov ( R 1 , R 2 ) = α 3 Heckman Factor Models: A Review of General Models

• ∴ We know σ 2 θ from Cov ( R 1 , R 2 ). • From Cov ( R 1 , R 3 ) we know α 3 , α 4 , α 5 . • Can get the variances of the ε i from variances of the R i Var ( R i ) = α 2 i σ 2 θ + σ 2 ε i . • If T = 2 , all we can identify is α 1 α 2 σ 2 θ . • If α 1 = 1 , σ 2 θ = 1 , we identify α 2 . • Otherwise model is fundamentally underidentified. Heckman Factor Models: A Review of General Models

2 Factors: (Some Examples) θ 1 ⊥ ⊥ θ 2 ε i ⊥ ⊥ ε j ∀ i � = j R 1 = α 11 θ 1 + (0) θ 2 + ε 1 R 2 = α 21 θ 1 + (0) θ 2 + ε 2 R 3 = α 31 θ 1 + α 32 θ 2 + ε 3 R 4 = α 41 θ 1 + α 42 θ 2 + ε 4 R 5 = α 51 θ 1 + α 52 θ 2 + ε 5 Let α 11 = 1, α 32 = 1. (Set scale) Heckman Factor Models: A Review of General Models

Cov ( R 1 , R 2 ) = α 21 σ 2 θ 1 Cov ( R 1 , R 3 ) = α 31 σ 2 θ 1 Cov ( R 2 , R 3 ) = α 21 α 31 σ 2 θ 1 • Form ratio of Cov ( R 2 , R 3 ) Cov ( R 1 , R 2 ) = α 31 , we identify ∴ α 31 , α 21 , σ 2 θ 1 , as before. Cov ( R 1 , R 4 ) = α 41 σ 2 ∴ since we know σ 2 θ 1 ∴ we get α 41 . θ 1 , . . . Cov ( R 1 , R k ) = α k 1 σ 2 θ 1 • ∴ we identify α k 1 for all k and σ 2 θ 1 . Heckman Factor Models: A Review of General Models

Cov ( R 3 , R 4 ) − α 31 α 41 σ 2 θ 1 = α 42 σ 2 θ 2 Cov ( R 3 , R 5 ) − α 31 α 51 σ 2 θ 1 = α 52 σ 2 θ 2 Cov ( R 4 , R 5 ) − α 41 α 51 σ 2 θ 1 = α 52 α 42 σ 2 θ 2 , • By same logic, Cov ( R 4 , R 5 ) − α 41 α 51 σ 2 θ 1 = α 52 Cov ( R 3 , R 4 ) − α 31 α 41 σ 2 θ 1 • ∴ get σ 2 θ 2 of “2” loadings. Heckman Factor Models: A Review of General Models

• If we have dedicated measurements on each factor do not need a normalization on the factors of R . • Dedicated measurements set the scales and make factor models interpretable: M 1 = θ 1 + ε 1 M M 2 = θ 2 + ε 2 M Cov ( R 1 , M ) = α 11 σ 2 θ 1 Cov ( R 2 , M ) = α 21 σ 2 θ 1 Cov ( R 3 , M ) = α 31 σ 2 θ 1 Cov ( R 1 , R 2 ) = α 11 α 12 σ 2 θ 1 , Cov ( R 1 , R 3 ) = α 11 α 13 σ 2 so we can identify α 12 σ 2 θ 1 , θ 1 • ∴ We can get α 12 , σ 2 θ 1 and the other parameters. Heckman Factor Models: A Review of General Models

General Case T × 1 = M R T × 1 + Λ K × 1 + ε T × K θ T × 1 • θ are factors, ε uniquenesses E ( ε ) = 0 σ 2  0 · · · 0  ε 1 . . σ 2 0 0 .   ε 2 Var ( εε ′ ) = D =   . . ... . .   . 0 .   σ 2 0 · · · 0 ε T E ( θ ) = 0 Var ( R ) = ΛΣ θ Λ ′ + D Σ θ = E ( θθ ′ ) Heckman Factor Models: A Review of General Models

• The only source of information on Λ and Σ θ is from the covariances. • (Each variance is “contaminated” by a uniqueness.) • Associated with each variance of R i is a σ 2 ε i . • Each uniqueness variance contributes one new parameter. • How many unique covariance terms do we have? • T ( T − 1) . 2 Heckman Factor Models: A Review of General Models

• We have T uniquenesses; TK elements of Λ. • K ( K − 1) elements of Σ θ . 2 • K ( K − 1) + TK parameters (Σ θ , Λ). 2 • Need this many covariances to identify model “ Ledermann Bound ”: T ( T − 1) ≥ TK + K ( K − 1) 2 2 Heckman Factor Models: A Review of General Models

Lack of Identification Up to Rotation • Observe that if we multiply Λ by an orthogonal matrix C , ( CC ′ = I ), we obtain Var ( R ) = Λ C [ C ′ Σ θ C ] C ′ Λ ′ + D • C is a “rotation.” • Cannot separate Λ C from Λ. • Model not identified against orthogonal transformations in the general case. Heckman Factor Models: A Review of General Models

Some common assumptions: (i) θ i ⊥ ⊥ θ j , ∀ i � = j σ 2  0 · · · 0  θ 1 . . σ 2 0 0 .   θ 2 Σ θ =   . . ... . .   . 0 .   σ 2 0 · · · 0 θ K Heckman Factor Models: A Review of General Models

joined with (ii)   1 0 0 0 · · · 0 0 0 0 · · · 0 α 21     1 0 0 · · · 0 α 31     α 41 α 42 0 0 · · · 0 Λ =     1 0 · · · 0 α 51 α 52     α 61 α 62 α 63 0 · · · 0   . . . .   . . . . . . . 1 . Heckman Factor Models: A Review of General Models

• We know that we can identify of the Λ , Σ θ parameters. K ( K − 1) ≤ T ( T − 1) + TK 2 2 # of free parameters data “Ledermann Bound” • Can get more information by looking at higher order moments. • (See, e.g., Bonhomme and Robin, 2009.) Heckman Factor Models: A Review of General Models

• Normalize: α I ∗ = 1, α 1 = 1 ∴ σ 2 ∴ α 1 . θ • Can make alternative normalizations. Heckman Factor Models: A Review of General Models

Recovering the Distributions Nonparametrically Theorem 1 Suppose that we have two random variables T 1 and T 2 that satisfy: T 1 = θ + v 1 T 2 = θ + v 2 with θ, v 1 , v 2 mutually statistically independent, E ( θ ) < ∞ , E ( v 1 ) = E ( v 2 ) = 0 , that the conditions for Fubini’s theorem are satisfied for each random variable, and the random variables possess nonvanishing (a.e.) characteristic functions, then the densities f ( θ ) , f ( v 1 ) , and f ( v 2 ) are identified. Proof. See Kotlarski (1967). Heckman Factor Models: A Review of General Models

• Suppose I = µ I ( X , Z ) + α I θ + ε I Y 0 = µ 0 ( X ) + α 0 θ + ε 0 Y 1 = µ 1 ( X ) + α 1 θ + ε 1 M = µ M ( X ) + θ + ε M . • System can be rewritten as I − µ I ( X , Z ) = θ + ε I α I α I Y 0 − µ 0 ( X ) = θ + ε 0 α 0 α 0 Y 1 − µ 1 ( X ) = θ + ε 1 α 1 α 1 M − µ M ( X ) = θ + ε M Heckman Factor Models: A Review of General Models

• Applying Kotlarski’s theorem, identify the densities of θ, ε I , ε 0 , ε 1 , ε M . α I α 0 α 1 • We know α I , α 0 and α 1 . • Can identify the densities of θ, ε I , ε 0 , ε 1 , ε M . • Recover the joint distribution of ( Y 1 , Y 0 ). � F ( Y 1 , Y 0 | X ) = F ( Y 1 , Y 0 | θ, X ) dF ( θ ) . • F ( θ ) is known. F ( Y 1 , Y 0 | θ, X ) = F ( Y 1 | θ, X ) F ( Y 0 | θ, X ) . • F ( Y 1 | θ, X ) and F ( Y 0 | θ, X ) identified F ( Y 1 | θ, X , S = 1) = F ( Y 1 | θ, X ) F ( Y 0 | θ, X , S = 0) = F ( Y 0 | θ, X ) . • Can identify the number of factors generating dependence among the Y 1 , Y 0 , C , S and M . Heckman Factor Models: A Review of General Models