Inference when identifying assumptions are doubted. A. Theory B. Applications

Transcription

1 Inference when identifying assumptions are doubted A. Theory B. Applications 1

2 A. Theory Structural model of interest: A y t B 1 y t1 B m y tm u t nn n1 u t i.i.d. N0, D D diagonal 2

3 Bayesian approach: Summarize whatever information we have that helps identify A in the form of a density pa. pa is highest for values of A we think are most plausible. pa 0 for values of A we rule out altogether. 3

4 pa could also impose sign restrictions, zeros, or assign small but nonzero probabilities to violations of these constraints. 4

5 Will use natural conjugate priors for other parameters: pd A i1 pd ii A 1 A i, i d ii Ed 1 ii A i / i Vard 1 2 ii A i / i uninformative priors: i, i 0 n 5

6 B B 1 B 2 B m n pb D, A i1 pb i D, A b i A, D Nm i, d ii M i uninformative priors: M i 1 0 6

7 Recommended default priors (Minnesota prior) Doan, Litterman, Sims (1984) Sims and Zha (1998) elements of m i corresponding to lag 1 given by a i all other elements of m i are zero M i diagonal with smaller values on bigger lags prior belief that each element of y t behaves like a random walk i function of A (or prior mode of pa) and scale of data 7

8 Likelihood: py T A, D, B 2 Tn/2 deta T D T/2 exp 1/2 T t1 Ay t Bx t1 D 1 Ay t Bx t1 prior: pa, D, B papd ApB A, D posterior: pa, D, B Y T py T A,D,BpA,D,B py T A,D,BpA,D,BdAdDdB pa Y T pd A, Y T pb A, D, Y T 8

9 Exact Bayesian posterior distribution (all T: b i A, D, Y T Nm i, d ii M i Y i 1Tk X i ktk a i y 1,..., a i y T, m i P i x 0 x T1 P i m i X i X i M i X i X i 1 1 X i y i P i P i M i 1 If uninformative prior (M i 1 0 then m i a i T 9

10 Frequentist interpretation of Bayesian posterior distribution as T : If prior on B is not dogmatic (that is, if M i 1 is finite), then m p i Ex t1 x t1 1 Ex t1 y t a i 0 a i M i p 0 b i A, D, Y T p 0 a i 10

11 d ii Posterior distribution for D A 1 A, Y T i T/2, i i /2 i Y i Y i Y i X i X i X i If M 1 i 0, i Ta i Ta i T T 1 T t1 t t, t y t x t1 ( t are unrestricted OLS residuals) 1 X i Y i 11

12 If priors on B and D are not dogmatic (that is, if M i 1, i, i are all finite) then i /T p a i 0 a i 0 Ey t x t1 Ey t x t1 Ex t x t 1 Ex t1 y t d ii A, Y T p ai 0 a i 12

13 Posterior distribution for A pa Y T k T padeta TA T/2 n 2i /T i1 i /T i T/2 k T constant that makes this integrate to 1 pa prior If M 1 i 0, and i i 0, pa Y T k TpA deta TA T/2 det diag(a TA T/2 13

14 pa Y T k TpA deta TA T/2 det diag(a TA If evaluated at A for which A TA diag(a TA, pa Y T k T pa T/2 14

15 pa Y T k TpA deta TA T/2 det diag(a TA Hadamard s Inequality: If evaluated at A for which A TA diag(a TA, det diag(a TA deta TA pa Y T 0 T/2 15

16 pa Y T kpa if A S 0 0 otherwise S 0 A: A 0 A diagonal 0 Ey t x t1 Ey t x t1 Ex t x t 1 Ex t1 y t 16

17 Special case: if model is point-identified (so that S consists of a single point), then posterior distribution converges to a point mass at true A 17

18 General case: The posterior distribution pa,d,b Y T summarizes uncertainty not just from finite data set but also our doubts about the identification itself. 18

19 Procedure: Draw A r from pa Y T Draw D r from pd A,Y T Draw B r from pb A, D Y T Repeat for r 1,...,

20 Nonorthogonalized impulse-response function: s y ts nn t Structural impulse-response function: H s y ts u nn t s A 1 20

21 Collect unknown elements of A, B, D in a vector and collect the row i, col j element S1 of the sequence of structural IRF H s s0 in ans1 vector h ij. Suppose we have a scalar-valued loss function gh ij,ĥ ij that specifies how much we lose if we announce an estimateĥ ij when the true value is h ij. From statistical decision theory the optimal estimate is ĥ ij arg gh ij,h ij p Y Td min h ij 21

22 Proposition 1 (Quadratic loss). Suppose gh ij,ĥ ij ĥ ij h ij Wĥ ij h ij for W positive definite S S. Then ĥ ij h ij p Y T d i.e., element-by-element posterior mean. Estimate by R 1 R r1 h ij r 22

23 Proposition 2 (absolute loss). Suppose gh ij,ĥ ij 0 h o 0 ij ĥ ij 1 h 1 1 ij ĥ ij S1 h S1 S1 ij ĥ ij Then optimal estimate h ij is posterior median (element-by-element). Calculate by median draw for h s ij r. 23

24 Historical decompositions: y ts ŷ s1 ts t m0 m tsm ŷ s1 ts t m0 m A 1 u tsm This decomposes value of y ts into forecast at t and the n structural shocks between t and ts. 24

25 Posterior mean or median of this magnitude gives optimal estimate and 95% posterior regions around this point summarize uncertainty from the data along with uncertainty about the model itself. 25

26 B. Applications Example 1: traditional Cholesky identification Kilian AER (2009) q t world oil production y t real global economic activity p t real price of oil 26

27 qy qp yp 0 oil supply: q t qy y t qp p t b 1 x t1 u 1t economic activity: y t yq q t yp p t b 2 x t1 u 2t inverse of oil demand curve: p t pq q t py y t b 3 x t1 u 3t 27

28 Bayesian translation: I put absolutely zero possibility on any A unless the (1,2), (1,3), and (2,3) elements are all zero. I have no information at all about the (2,1), (3,1), and (3,2) elements. 28

29 Blue: posterior median IRF as calculated using Baumeister- Hamilton algorithm for dogmatic prior. Red: IRF calculated using Kilian (AER, 2009) Cholesky. 29

30 Posterior density of short-run demand elasticity Implied by Bayesian interpretation of Kilian (AER, % posterior probability that demand elasticity > 0. 94% posterior probability that abs(elasticity) > 2. 30

31 Example 2: Labor demand and supply demand: n t k d d w t b d 11 w t1 b d 12 n t1 b d 21 w t2 b d 22 n t2 b d m1 w tm b d d m2 n tm u t supply: n t k s s w t b s 11 w t1 b s 12 n t1 b s 21 w t2 b s s s s 22 n t2 b m1 w tm b m2 n tm u t 31

32 Maximum likelihood estimate of α and β 5 The function β(α) h L h H 0 β α 32

33 5 Contours for log likelihood h H 0 β α 33

34 What do we know from other sources about absolute value of short-run wage elasticity of labor demand? Hamermesh (1996) survey of microeconometric studies: 0.1 to 0.75 Lichter, et. al. (2014) meta-analysis of 942 estimates: lower end of Hamermesh range Theoretical macro models can imply value above 2.5 (Akerlof and Dickens, 2007; Gali, et. al. 2012) 34

35 Prior for : Student t with location c, scale, d.f., truncated by 0 c 0. 6, 0. 6, 3 Prob Prob

36 Student t prior for labor demand elasticity 36

37 What do we know from other sources about wage elasticity of labor supply? Long run: often assumed to be zero because income and substitution effects cancel (e.g., Kydland and Prescott, 1982) Short run: often interpreted as Frisch elasticity Reichling and Whalen survey of microeconometric studies: Chetty, et. al. (2013) review of 15 quasiexperimental studies: < 0.5 Macro models often assume value greater than 2 (Kydland and Prescott, 1982, Cho and Cooley, 1994, Smets and Wouters, 2007) 37

38 Prior for : Student t with location c, scale, d.f. a, truncated by 0 c 0. 6, 0. 6, 3 Prob Prob

39 Student t prior for labor supply elasticity 39

40 0 Contours for log of prior -1-2 β α Contours for log of posterior -1-2 β α 40

41 Could we also use information about longrun labor supply elasticity? y t w t,n t (data used for y t in VAR as estimated) y t w t, n t (data in levels) u t u d t, u s t (vector of structural shocks) 41

42 y ts u t y ts t t u t s A L 2 L 2 y ts u t I n 1 L 2 L 2 m L m 1 y ts u t y ts1 u t y t u t s A 1 s1 A 1 0 A 1 42

43 y ts u t s A 1 s1 A 1 0 A 1 lim y ts s u A 1 t I n 1 2 m 1 A 1 AI n 1 2 m 1 A B 1 B 2 B m 1 43

44 lim y ts s u A B 1 B 2 B m 1 t Labor demand shock (shock #1) has zero long run effect on employment (second element of y ts if and only if 2, 1 element is zero: 0 s s s b 11 b 21 s b m1 44

45 0 s s s s b 11 b 21 b m1 Usual approach: impose this condition as untestable identifying assumption Our suggestion: instead represent as prior belief, s s s b 11 b 21 b m1 A, D N s, d 22 V V 0. 1 prior given same weight as 10 observations on y t 45

46 Prior and posterior distributions for short-run elasticities and long-run impact 1 β d 6 α s α s + b s + b s s b m

47 Posterior medians and 95% credibility regions for structural impulse-response functions Labor Demand Shock Labor Supply Shock 3 Real wage 1 Real wage Percent 2 1 Percent Quarters Quarters 6 Employment 6 Employment 5 5 Percent Percent Quarters Quarters 47

48 Response of employment to labor demand shock α s Percent V = V = Percent V = V = Percent V = V = Percent V = V = Quarters

49 Example 3: effects of monetary policy y t output gap t inflation (year-over-year PCE) r t fed funds rate t 1986:Q1-2008:Q3 49

50 Aggregate Supply or Phillips Curve y t k s s t b s s x t1 u t Aggregate Demand or Euler Equation y t k d d t d r t b d d x t1 u t Monetary Policy or Taylor Rule r t k m y y t t b m m x t1 u t 50

51 Commonly used Taylor Rule r t r 1 y y t 1 t m r t1 ru t is a special case of our equation r t k m y y t t b m m x t1 u t y 1 y 1 51

52 Bayesian prior: y Student t location 0.5, scale 0.4, d.f. 3 truncated to be positive Prob( y , Prob( y Student t1. 5,0.4,3 Beta2. 6,2.6 mean 0.5, std dev

53 A 1 s 0 1 d d 1 y

54 Commonly used dynamic IS curve y t b y d y t1 t r t t1 t u t intertemporal elasticity of substitution DSGE would imply y t1 t c y y x t t1 t c x t 54

55 y t1 t c y y x t t1 t c x t One approach: find DSGE-implied values for y and in terms of deep structural parameters, use these for prior. Our approach: use prior beliefs about reduced-form directly. 55

56 Minnesota prior: the most useful variable for predicting any variable is its own lagged value. y x t y y t x t t Minnesota prior: everything is a random walk y 1 For our variables (output gap, inflation) we instead expect y

57 y t b y d y t1 t r t t1 t u t b y y y t r t d t u t y t b y /1 y r t /1 y d t ũ t Intertemporal elasticity of substitution 0.5 2/3 57

58 So for our AD equation y t k d d t d r t b d xt1 u t d we expect d /1 y 0.5/10.752/3 1 d /1 y Bayesian prior d Student t1,0.4,3 truncated 0 d Student t0.75,0.4,3 no sign restriction 58

59 Phillips Curve y t k s s t b s x t1 u t s s Student t2,0.4, 3 truncated 0 59

60 Priors for Impacts of Shocks H A 1 1 deta H deta s 1 d 1 y d d 1 H d d 1 s s d d 1 y 1 1 d 1 d y 1 s y s d 60

61 Priors for Impacts of Shocks Sign restrictions on contemporaneous coefficients s 0, d 0, y 0, 0, and1 0 imply the following signs on H : signh??? 61

62 We could use prior information about the other signs either dogmatically or with incomplete confidence. We implement the latter using a new distribution we call the asymmetric t distribution. 62

63 Let vx be density of standard Student t with degrees of freedom evaluated at the point x. vx 1/2 1/2 /2 Matlab: tpdf(x,nu) 1 x2 Let x be N0,1 cdf 1/2 63

64 Consider density ph k 1 h v h h h / h h h/ h h 0 h h/ h 1/2 h h Student t( h, h, h h 0 gives positive skew h 0 gives negative skew 64

65 ph k h 1 v h h h / h h h/ h h h h/ h 0 for h 0 1 for h 0 h Student t( h, h, h truncated h 0 dogmatic prior that h 0 h h h/ h 1 for h 0 0 for h 0 h Student t( h, h, h truncated h 0 dogmatic prior that h 0 65

66 A favorable supply shock will raise output and lower inflation in equilibrium if h 1 d d 1 0 Prior for h 1 asymmetric Student t with h 0.1, h1 1 (from simulating draws from p h1 3, h1 4 66

67 Prior for h 67

68 signh??? A monetary contraction that raises fed funds rate 1% in equilibrium will change output gap by h 2 s d s d (expect 0 prior: h 2 asymmetric t0. 3, 0. 5, 3,2. 68

69 Prior for h 69

70 Overall prior for A: logpa log p s log p d log p d log p y log p log p log ph 1 d, d,, log ph 2 s, d, d 70

71 Priors for Structural Variances d 1 ii A,a i Sa i 2 S is the sample variance matrix of univariate residuals from AR(4): T s ij T 1 t1 ê it ê jt 71

72 Priors for Lagged Structural Coefficients Minnesota prior: coeff on own lag in reduced form is 0.75 first three elements of b i may be close to 0.75a i for a i the i th column of A all other coeffs 0 Also have information that third element of b r should be near b i A,D Nm i A,d ii M i 72

73 Prior (red) and posterior (blue) distributions for contemporaneous coefficients 73

74 Impulse-response functions 74 Solid blue lines: posterior median. Shaded regions: 68% posterior credibility set. Dotted blue lines: 95% posterior credibility set. Dashed red lines: prior median.

75 Prior and posterior probabilities that effect of shock is positive at horizon s Variable Supply shock Demand shock Monetary policy shock (1) (2) (3) (4) (5) (6) Prior Posterior Prior Posterior Prior Posterior s = 0 y π r s = 1 y π r s = 2 y π r

76 Historical decomposition of the output gap Dashed red: actual data in deviation from mean. Solid blue: portion attributed to indicated structural shock. Shaded regions: 68% posterior credibility sets. Dotted blue: 95% posterior credibility sets. 76

77 Historical decomposition of inflation Dashed red: actual data in deviation from mean. Solid blue: portion attributed to indicated structural shock. Shaded regions: 68% posterior credibility sets. Dotted blue: 95% posterior credibility sets. 77

78 Historical decomposition of fed funds rate Dashed red: actual data in deviation from mean. Solid blue: portion attributed to indicated structural shock. Shaded regions: 68% posterior credibility sets. Dotted blue: 95% posterior credibility sets. 78

79 Because we have used multiple sources of prior information, we can look at what difference it makes if we drop any one. E.g., replace d Student t0.75,0.4,3 with d Student t0.75,10,3. 79

80 Plot of Student t density with location parameter 0.75, 3 degrees of freedom, and scale parameter of 0.4, 2, or

81 Response of output gap to monetary shock with an uninformative prior for indicated parameter. Solid blue: posterior median. Dashed red lines: benchmark posterior. 81