Introduction to Bayesian Estimation Wouter J. Den Haan London - PowerPoint PPT Presentation

Introduction to Bayesian Estimation Wouter J. Den Haan London School of Economics � 2011 by Wouter J. Den Haan c May 31, 2015

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Overview • Maximum Likelihood • A very useful tool: Kalman filter • Estimating DSGEs • Maximum Likelihood & DSGEs • formulating the likelihood • Singularity when #shocks ≤ number of observables • Bayesian estimation • Tools: • Metropolis Hastings

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Standard Maximum Likelihood problem Theory: y t = a 0 + a 1 x t + ε t N ( 0, σ 2 ) ε t ∼ x t : exogenous Data: { y t , x t } T t = 1

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other ML estimator T ∏ max p ( ε t ) a 0 , a 1 , σ t = 1 where ε t = y t − a 0 − a 1 x t � − ε 2 � 1 t p ( ε t ) = √ exp 2 σ 2 2 π σ

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other ML estimator � � − ( y t − a 0 − a 1 x t ) 2 T 1 ∏ √ max exp 2 σ 2 a 0 , a 1 , σ σ 2 π t = 1

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Rudolph E. Kalman born in Budapest, Hungary, on May 19, 1930

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Kalman filter • Linear projection • Linear projection with orthogonal regressors • Kalman filter The slides for the Kalman filter is based on Ljungqvist and Sargent’s textbook

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Linear projection • y : n y × 1 vector of random variables • x : n x × 1 vector of random variables • First and second moments exist x � = Σ xx E y = µ y y = y − µ y ˜ E ˜ x ˜ y � = Σ yy E x = µ x x = x − µ x ˜ E ˜ y ˜ x � = Σ yx E ˜ y ˜

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Definition of linear projection The linear projection of y on x is the function � E [ y | x ] = a + Bx , a and B are chosen to minimize � ( y − a + Bx )( y − a + Bx ) � � E trace

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Formula for linear projection The linear projection of y on x is given by � E [ y | x ] = µ y + Σ yx Σ − 1 xx ( x − µ x )

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Difference with linear regression problem • True model: Bx + ¯ ¯ y = Dz + ε , E x = E z = E ε = 0, E [ ε | x , z ] = 0 , E [ z | x ] � = 0 ¯ B : measures the effect of x on y keeping all else–also z and ε –constant. • Particular regression model: y = ¯ Bx + u

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Difference with linear regression problem Comments: • Least-squares estimate � = ¯ B • Projection: � D � E [ y | x ] = Bx = ¯ Bx + ¯ E [ z | x ] • Projection well defined linear projection can include more than the direct effect:

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Message: • You can always define the linear projection • you don’t have to worry about the properties of the error term.

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Linear Projection with orthogonal regressors • x = [ x 1 , x 2 ] and suppose that Σ x 1 x 2 = 0 • x 1 and x 2 could be vectors � µ y + Σ yx Σ − 1 E [ y | x ] = xx ( x − µ x ) � � Σ − 1 � � 0 x 1 x 1 Σ yx 1 Σ yx 2 = µ y + ( x − µ x ) Σ − 1 0 x 2 x 2 µ y + Σ y x 1 Σ − 1 x 1 x 1 ( x 1 − µ x 1 ) + Σ y x 2 Σ − 1 = x 2 x 2 ( x 2 − µ x 2 ) Thus � E [ y | x ] = � E [ y | x 1 ] + � E [ y | x 2 ] − µ y (1)

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Time Series Model x t + 1 = Ax t + Gw 1, t + 1 y t = Cx t + w 2, t Ew 1, t = Ew 2, t = 0 � w 1, t + 1 � � w 1, t + 1 � � � V 1 � V 3 E = V � w 2, t w 2, t V 2 3

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Time Series Model • y t is observed, but x t is not • the coefficients are known (could even be time-varying) • Initial condition: • x 1 is a random variable (mean µ x 1 & covariance matrix Σ 1 ) (it is not unusual that x t is simply set equal to µ x 1 . • w 1, t + 1 and w 2, t are serially uncorrelated and orthogonal to x 1

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Objective The objective is to calculate � � E t x t + 1 ≡ E [ x t + 1 | y t , y t − 1 , · · · , y 1 , ˜ x 1 ] � � � x t + 1 | Y t , ˜ = E x 1 where ˜ x 1 is an initial estimate of x 1 Trick: get a recursive formulation

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Orthogonalization of the information set • Let y t = y t − � • ˆ E [ y t | ˆ y t − 1 , ˆ y t − 2 , · · · , ˆ y 1 , ˜ x 1 ] Y t = { ˆ • ˆ y t , ˆ y t − 1 , · · · , ˆ y 1 } x 1 , ˆ Y t } = space spanned by { ˜ • space spanned by { ˜ x 1 , Y t } • That is, anything that can be expressed as a linear x 1 , ˆ Y t } can be expressed as a combination with elements in { ˜ linear combination of elements in { ˜ x 1 , Y t } .

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Orthogonalization of the information set • Then � � = � � � = C � � � � y t + 1 | Y t , ˜ y t + 1 | ˆ Y t , ˜ x t + 1 | ˆ Y t , ˜ E x 1 E x 1 E x 1 (2)

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Derivation of the Kalman filter From (1) we get � � � � = � � y t ] + � Y t − 1 , ˜ x t + 1 | ˆ Y t , ˜ x t + 1 | ˆ E x 1 E [ x t + 1 | ˆ E x 1 − E x t + 1 (3) The first term in (3) is a standard linear projection: y t )] − 1 ( ˆ � E [ x t + 1 | ˆ E x t + 1 + cov ( x t + 1 , ˆ y t ) [ cov ( ˆ y t , ˆ y t − E ˆ y t ] = y t ) y t )] − 1 ˆ = E x t + 1 + cov ( x t + 1 , ˆ y t ) [ cov ( ˆ y t , ˆ y t

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Some algebra • Similar to the definition of ˆ y t , let x t + 1 − � x t + 1 ˆ = E [ x t + 1 | ˆ y t , ˆ y t − 1 , · · · , ˆ y 1 , ˜ x 1 ] x t + 1 − � = E t x t + 1 x � • Let Σ ˆ x t = E ˆ x t ˆ t x t C � + GV 3 cov ( x t + 1 , ˆ y t ) = A Σ ˆ x t C � + V 2 cov ( ˆ y t , ˆ y t ) = C Σ ˆ • To go from unconditional covariance, cov ( · ) , to conditional Σ ˆ x t requires some algebra (see appendix of Ljungqvist-Sargent for details)

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Using the derived expressions � E [ x t + 1 | ˆ y t ] y t )] − 1 ˆ = E x t + 1 + cov ( x t + 1 , ˆ y t ) [ cov ( ˆ y t , ˆ y t � � � � − 1 ˆ x t C � + GV 3 x t C � + V 2 = E x t + 1 + A Σ ˆ C Σ ˆ y t (4)

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Derivation Kalman filter • Now get an expression for the second term in (3). • From x t + 1 = Ax t + Gw 1, t + 1 , we get � � � � � x t + 1 | ˆ Y t − 1 , ˜ = A � x t | ˆ Y t − 1 , ˜ = A � E x 1 E x 1 E t − 1 x t (5)

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Using (4) and (5) in (3) gives the recursive expression � E t x t + 1 = A � E t − 1 x t + K t ˆ y t where � � � � − 1 x t C � + GV 3 x t C � + V 2 K t = A Σ ˆ C Σ ˆ

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Prediction for observable From y t + 1 = Cx t + 1 + w 2, t + 1 we get � x 1 ] = C � E [ y t + 1 | Y t , ˜ E t x t + 1 Thus y t + 1 = y t + 1 − C � ˆ E t x t + 1

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Updating the covariance matrix • We still need an equation to update Σ ˆ x t . This is actually not that hard. The result is x t A � + GV 1 G � − K t ( A Σ ˆ x t C � + GV 3 ) � Σ ˆ x t + 1 = A Σ ˆ • Expression is deterministic and does not depend particular realizations. That is, precision only depends on the coefficients of the time series model

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Applications Kalman filter • signal extraction problems • GPS, computer vision applications, missiles • prediction • simple alternative to calculating inverse policy functions • (see below)

Overview ML Kalman Filter Estimating DSGEs ML & DSGE Bayesian estimation MCMC Other Estimating DSGE models • Forget the Kalman filter for now, we will not use it for a while • What is next? • Specify the neoclassical model that will be used as an example • Specify the linearized version • Specify the estimation problem • Maximum Likelihood estimation • Explain why Kalman filter is useful • Bayesian estimation • MCMC, a necessary tool to do Bayesian estimation