Filtering with limited information Thorsten Drautzburg, 1 Jes - PowerPoint PPT Presentation

Filtering with limited information Thorsten Drautzburg, 1 Jes´ andez-Villaverde, 2 and Pablo Guerr´ on 3 us Fern´ January 25, 2018 2 Federal Reserve Bank of Philadelphia 2 University of Pennsylvania 3 Boston College

Lars Peter Hansen (2014) Hansen (1982) builds on a long tradition in econometrics of ‘doing something without having to do everything.’ This entails the study of partially specified models, that is, models in which only a subset of economic relations are formally delineated. 1

Motivation • Filtering shocks from a dynamic model is a central task in macroeconomics: 1. Path of shocks is a reality check for the model. 2. Historical decompositions and counterfactuals. 3. Forecasting and optimal policy recommendations when laws of motions are state-dependent. 4. Structural estimation. • However, filtering is hard, and there is no universal and easy-to-apply algorithm to implement it. 2

Alternatives • A possibility ⇒ sequential Monte Carlos: Fern´ andez-Villaverde, Rubio-Ram´ ırez, and Schorfheide (2016). 1. Specification of the full model, including auxiliary assumptions. 2. Computationally costly. 3. Curse of dimensionality. • Some of the previous points also hold for even the simple linear, Gaussian case where we can apply the Kalman filter. • Can we follow Hansen’s suggestion and ‘do something without having to do everything”? ⇒ Yes! • Partial information filter. 3

Environment • Model: f ( x t , y t , E t [ g ( x t +1 , y t +1 , x t , y t )]) = 0 • Deterministic steady state: f (¯ x , ¯ y , g (¯ x , ¯ y , ¯ x , ¯ y )]) = 0 • Estimation: x t , y t , � f ( � E t [ g ( � x t +1 , y t +1 , � x t , y t )]) = 0 4

Factorization • Factorization of g ( ◦ ) g ( x t +1 , y t +1 , x t , y t ) ≡ g 1 ( x t +1 , y t +1 , x t , y t ) × g 2 ( x t +1 , y t +1 , x t , y t ) • Then: E t [ g ( x t +1 , y t +1 , x t , y t ) i ] ≡ E t [ g 1 ( x t +1 , y t +1 , x t , y t ) i ] × E t [ g 2 ( x t +1 , y t +1 , x t , y t ) i ] + Cov t [ g 1 ( x t +1 , y t +1 , x t , y t ) i , g 2 ( x t +1 , y t +1 , x t , y t ) i ] • We need to approximate conditional first and second moments: 1. Equilibrium conditions of the model. 2. Observed expectations. 3. Auxiliary statistical model. 5

Auxiliary statistical model • VAR(1) in g 1 , t , g 2 , t , (a subset of n � y elements of) y t , and � x t for t = 1 , ..., T . • Why? • If we collect variables collected in ξ t : ξ t = µ + A ξ t − 1 + ǫ t , Var[ ǫ t ] = Σ • Ordering g 1 , t and g 2 , t as the first two variables of the VAR, we can write: E t [ g ( x t +1 , y t +1 , x t , y t )] i ≡ ( µ i + e ′ i A ξ t )( µ i + m + e ′ i + m A ξ t ) + Σ i , i + m where e i is a selection vector. • Thus: f ( x t , y t , ( µ i + e ′ i A ξ t )( µ i + m + e ′ i + m A ξ t ) + Σ i , i + m ) i = 0 6

Two approaches x T (and associated parameters for the VAR) that solve • (Fixed point): we find � for all i : f ( x t , y t , ( µ i + e ′ i A ξ t )( µ i + m + e ′ i + m A ξ t ) + Σ i , i + m ) i = 0 x ∀ t and A (0) = 0 , µ (0) = ¯ In practice, initialize � E (0) x (0) [ g ( ◦ )] based on � = ¯ ξ T t t and Σ (0) = 0 and iterate until convergence. • (Gibbs sampler): From d = 1 , . . . , D , iterate on: t =1 , µ ( d ) , A ( d ) , Σ ( d ) ∼ P ( µ, A , Σ | ξ t ). x ( d ) } T 1. Given { � t x ( d +1) 2. Given y T , µ ( d ) , A ( d ) , Σ ( d ) , solve for { � } T t =1 for all i in: t f ( x t , y t , ( µ i + e ′ i A ξ t )( µ i + m + e ′ i + m A ξ t ) + Σ i , i + m ) i = 0 x (0) Start the Gibbs sampler from � = ¯ x or the fixed point above. t The Gibbs sampler allows us to quantify estimation uncertainty. 7

Tobin’s Q • Representative household � s � ∞ � � max β t − 1+ s u ( c t + s , n t + s ) { c t , n t , i t } E s =0 u =1 s.t. � � i t � 2 � 1 − χ k t +1 = (1 − δ t ) k t + e ξ t − (1 + g ) i t 2 i t − 1 c t + i t = e z t k α t − 1 n 1 − α t • Four shocks: β t , δ t , ξ t , z t follow log-linear AR(1) processes 8

Solution (I) ) 1 − η = λ t (1 − κ (1 − η ) n 1+1 /φ c − η t t � k t − 1 � α c 1 − η (1 − κ (1 − η ) n 1+1 /φ ) − η (1 + 1 /φ ) κ (1 − η ) n 1 /φ = λ t (1 − α ) e z t t t t n t � � i t � 2 � i t � i t � 1 − χ λ t = µ t e ξ t − (1 + g ) − χ − (1 + g ) 2 i t − 1 i t − 1 i t − 1 � � i t +1 �� i t +1 � 2 � + β t E t µ t +1 χ − (1 + g ) i t i t � � n t +1 � 1 − α � (1 − δ t +1 ) µ t +1 + λ t +1 α e z t +1 µ t = β t E t k t 9

Solution (II) ) 1 − η = λ t (1 − κ (1 − η ) n 1+1 /φ c − η t t � k t − 1 � α c 1 − η (1 − κ (1 − η ) n 1+1 /φ ) − η (1 + 1 /φ ) κ (1 − η ) n 1 /φ = λ t (1 − α ) e z t t t t n t � � i t � 2 � i t � i t � 1 − χ 1 = q t e ξ t − (1 + g ) − χ − (1 + g ) 2 i t − 1 i t − 1 i t − 1 � � i t +1 �� i t +1 � 2 � λ t +1 + E t β t q t +1 χ − (1 + g ) λ t i t i t � � �� λ t +1 (1 − δ t +1 ) q t +1 + r k q t = E t β t t +1 λ t � � 1 − α n t +1 where r k t ≡ α e z t +1 k t 10

Quantitative set-up • Calibration: n = 1 1. ¯ 3 . 2. ¯ g = 0 . 5%, a 2% annual growth rate. 3. η = 1 . 0, to have separable preferences for now. 4. φ = 1, a typical value for the Frisch elasticity. 5. ¯ δ = 2%, implying an 8% annual depreciation rate. 6. β = 0 . 995, implying a annualized real rate of about 2% + 4 η ¯ g = 5%. 7. ρ z = 0 . 95, σ z = 0 . 76%. 8. ρ β = 0 . 8, σ β = 1 . 0%. 9. ρ ξ = 0 . 9, σ ξ = 1 . 0%. 10. ρ δ = 0 . 75, σ δ = 1 . 0%. • Solved the model using 3rd order perturbation methods with pruning, as in Andreasen, Fern´ andez-Villaverde, and Rubio Ram´ ırez (2017). • Simulated data: We simulate the model for 2,000 periods after a burn-in of 1,000 periods. 11

Filter set up λ t +1 • We want to filter a single variable, q t using data on the SDF M t +1 ≡ β t λ t , the rental rate on capital r k t , and the risk-free rate r f t . • Set x t = q t , y t = [ � M t , r k t , r f t ], and rewrite the conditional expectation of the return on investment as: � � (1 − ¯ g ( x t +1 , y t +1 , x t , y t ) ≡ y 1 , t +1 × δ ) x t +1 + y 2 , t +1 � � = � (1 − ¯ δ ) q t +1 + r k M t +1 × t +1 with � M t +1 = ¯ β c t c t +1 . • Note use of misspecified model. • Thus � � f ( x t , y t , E t [ g ( x t +1 , y t +1 , x t , y t ]) ≡ − x t + E t g ( x t +1 , y t +1 , x t , y t ) � �� � � (1 − ¯ δ ) q t +1 + r k = − q t + E t M t +1 × t +1 12

Auxiliary statistical model • Using that, in equilibrium, 1 + r f t ≡ E t [ M t +1 ] − 1 , we can re-write previous expression as an VAR approximation: f ( x ( d ) , y t , � E ( d ) [ g ( x ( d − 1) , y t +1 , x ( d − 1) , y t ]) ≈ t t t +1 t − x ( d − 1) δ )Σ ( d ) 1 , 4 + Σ ( d ) + (1 − ¯ t 1 , 2 � � t ) − 1 × ( µ ( d ) + A ( d ) X ( d − 1) (1 − ¯ +(1 + r f δ ) e 4 + e 1 ) t • We already are setting up notation for Gibbs sampler. • For comparison purposes, we will also run: 1. A naive guess for q computed as q t = (1 − ¯ δ )+ mpk t +1 . 1+ r f t 2. A Kalman filter and smoother. 13

VAR • VAR(1) in ξ ( d ) q ( d ) ≡ [ M t +1 , r k t +1 , r f t +1 , � ]: t t q ( fp ) , T 1. Fixed point: We solve for a fixed point in the 2,000-dimensional vector � µ ( fp ) , A ( fp ) , and Σ ( fp ) . and the VAR parameters � 2. Gibbs sampler: 2.1 Set ξ ( d ) = [ M 0 , r k 0 , r f 0 , ¯ q ] 0 2.2 We sample Σ ( d ) and β ( d ) = [vec( A ( d ) ) ′ , ( µ ( d ) ) ′ ] ′ from Σ ( d ) | ξ ( d ) , T ∼ IW ( T − 1 , � Σ ( d ) OLS × T ) and β ( d ) | ξ ( d ) , T , Σ ( d ) ∼ N ( � β ( d ) OLS , Σ ( d ) ⊗ ((¯ ξ ( d − 1) ) ′ ¯ ξ ( d − 1) ) − 1 ), conditioning on ξ ( d ) = [ M 0 , r k 0 , r f 0 , ¯ q ]. 0 Σ ( d ) 2.3 We use a flat prior and define � OLS × T as the OLS sum of squared residuals and β ( d ) � OLS the OLS estimator/MLE of the coefficients. ¯ ξ is a T × 5 matrix with rows ¯ ξ t = [ ξ t − 1 , 1]. q ( d ) 14 2.4 We solve optimality condition for � . t

Kalman Filter: Kalman Smoother: corr=0.97, rel sd=0.49 corr=0.99, rel sd=1.20 0.2 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0.05 smoothed q filtered q 0 0 -0.05 -0.05 -0.1 -0.1 -0.15 -0.15 -0.2 -0.2 -0.25 -0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2 simulated q simulated q 15

Naive initial estimate Bayes full sample Fixed point corr=0.97, rel sd=0.33 corr=0.89, rel sd=1.35 corr=0.79, rel sd=0.72 0.2 0.4 0.2 0.15 0.15 0.3 0.1 0.1 0.2 0.05 0.05 filtered q filtered q initial q 0 0.1 0 -0.05 -0.05 0 -0.1 -0.1 -0.1 -0.15 -0.15 -0.2 -0.2 -0.2 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 simulated q simulated q simulated q 16

0.4 Partial filter: Posterior 0.3 Partial filter: Fixed point Truth 0.2 0.1 log q 0 -0.1 -0.2 -0.3 0 50 100 150 200 17

Political distribution risk and aggregate fluctuations Budd (2012), ‘Labor Relations – Striking a Balance,’ 4th ed. A popular framework for thinking about labor law is to consider a pendulum that can range from strong bargaining power for labor . . . to strong bargaining power for companies . . . . We stress the role that changes in 1. statutory labor law (including executive orders), 2. case law (courts and NLRB), and 3. political climate have on business cycles, income shares, and asset prices. 18