Bayesian DSGE Model Estimation

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1 Bayesian DSGE Model Estimation Lecture six Alexander Kriwoluzky Universiteit van Amsterdam February 12th, 2010 Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 1

2 Motivation We understand Bayesian DSGE model estimation now. It is important to check whether the estimation results are sensible. Check the posterior distribution. Compare it to the prior distribution. Plot the observable variables against the data. Plot the estimated structural shocks. Check the convergence of the chain. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 2

3 Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 3

4 Aim of week 6 Produce plots. Diagnose the MCM chain(s). Compute smoothed estimates. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 4

5 Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 5

6 Prior vs. Posterior plots Draw from the prior distribution. Form a matrix PP plot including the draws from the posterior distribution. Plot a histogram using the hist and bar commands: [NDPB, Cent] = hist(pp plot, numbins) bar(cent, NDPB numberdraws ) For many parameters use the subplot command. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 6

7 Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 7

8 Kalman filter revisited So far we have computed: A sequence of forecasted states {x t+1 t } T 1 t=0 = {E t[x t+1 ]} T 1 t=0 and associated MSE {Σ x,t+1 t } T 1 t=0. A sequence of forecast errors {s t } T t=1. Now we want to compute a sequence summarizing our knowledge about x: {x t } T t=1. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 8

9 Prerequisites Kalman smoothing Compute {x t t } T t=1 and {Σ x,t t} T t=1 using: x t t = x t t 1 + Σ t t 1 H (HΣ t t 1 H + Σ u ) 1 s t (1) Σ x,t t = Σ t t 1 Σ t t 1 H (HΣ t t 1 H + Σ u ) 1 HΣ t t 1 (2) Compute a sequence of the matrix {J t } T 1 t=1 from: J t = Σ x,t t F Σ 1 x,t+1 t (3) Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 9

10 Smoothed estimates The sequence of smoothed state variables {x t T } T t=1 is computed from: x t T = x t t + J t (x t+1 T x t+1 t ) (4) The knowledge of the state at time t given all the information at t (x t t ) is updated using knowledge of x t+1. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 10

11 Plot of a time series You want to plot the time series YY. time = 1 : number observation size plotx = ceil(sqrt(number observable )); size ploty = (sqrt(number observable )); Initialize figure(1). Take a for-loop (i) over the observable variables: subplot(size ploty, size plotx, i); plot(time, YY(:, i)) title([names observable (i, :)]); Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 11

12 Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 12

13 Issues Convergence Did the chain converged to the target distribution? Is the chain still influenced by the starting value? Within sequence correlation Do the draws really constitute independent draws? Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 13

14 Solutions Employ efficient acceptance rate. Discard early iterations of each sequence. (Brooks, Gelman: 50 %) Simulate multiple sequences with different starting points. Compare variation within and between sequences until the within variation roughly equals between variation. Compute the effective number of independent draws. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 14

15 Univariate convergence diagnostics Check whether moments of the parameter µ i : are similar within one sequence, are similar between sequences, can be estimated more precisely by increasing the size of the sequence. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 15

16 Within and between chain variance Denote µ i,j with i = 1... n and j = 1... m, where n is the length of the chain, and m the number of chains. The measures to be introduced rely on the normality assumption (approximately) for the parameter. The within-variance W is defined as: W = 1 m m j=1 s 2 j, s2 j = 1 n 1 The between-variance B is defined as: n (µ i,j µ.,j ) (5) i=1 B = n m 1 m ( µ.j µ.. ) 2, j=1 µ.j = 1 n n µ i,j, µ.. = 1 m i=1 m j=1 µ.j (6) Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 16

17 Scale reduction factor Marginal posterior variance of the estimand: Scale reduction factor V = n 1 n W + 1 n B (7) ˆR 2 = V W (8) Flaws: While W underestimates the variance, V overestimates the variance. It is necessary to correct ˆR 2. Normality assumption. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 17

18 Alternative ˆR 2 ˆR Interval is defined as the ratio of: 1. The length of total sequence interval Pool all n m draws. Take the 100( α Int )% and 100(1 α Int )% 2 2 interval. 2. Mean length of the within sequence intervals. For every j = 1... m compute the 100( α Int )% and 2 100(1 α Int )%. 2 Compute the mean. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 18

19 Scale reduction factor for higher moments For higher moments the scale reduction factor is given by: ˆR s = 1 m mn 1 j=1 1 m m(n 1) j=1 n i=1 µ ij µ.. s n i=1 µ ij µ.j s (9) Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 19

20 Number of effective draws Effective number of independent draws is given by: n eff = mn V B (10) Interpretation: If the draws are independent B is an unbiased estimate of V we have mn independent draws. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 20

21 Implementation I To give credits: The following implementation follows quite closely the one by the Dynare team. Define a matrix DIAGNOSTIC with dimension number of diagnostic iteration steps times 6 times number of parameters. For every parameter: Sort all m n draws in a matrix DIAGDRAWS. The matrix should have three columns: the value of the parameter, the number of the chain, the number of the draw. For the rolling sequence find the mean of the pooled sequence, of each sequence as well as the intervals. Hint: The rolling sequence is found by the following command: roll = DIAGDRAWS(find((DIAGDRAWS(:, 3) >= lowerbound)and(diagdraws(:, 3) <= upperbound)), 1 : 2) Hint II: The draws corresponding to the chain i = 1... n are found by: chaindraws = DIAGDRAWS(find(DIAGDRAWS(:, 2) == i)). Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 21

22 Implementation II The six columns of DIAGDRAWS are computed by: 1. roll(uppervalueinterval) roll(lowervalueinterval) 2. sum over all m: chaindraws(uppervalueinterval) chaindraws(lowervalueinterval) 3. Equation (9) numerator for s = Equation (9) denominator for s = Equation (9) numerator for s = Equation (9) denominator for s = 3. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 22

23 Homework Program the following functions in Matlab: 1. Plot prior and posterior distribution. 2. Assess convergence of your MCMC plot the univariate R for the estimated parameters. 3. Plot the smoothed shocks and variables. Also: plot the smoothed observable variables against the simulated observations. Aim of the week Prior vs. Posterior Kalman smoother Chain Diagnostics 23

Bayesian Inference for DSGE Models. Lawrence J. Christiano

Bayesian Inference for DSGE Models Lawrence J. Christiano Outline State space-observer form. convenient for model estimation and many other things. Bayesian inference Bayes rule. Monte Carlo integation.