New meeting times for Econ 210D Mondays 8:00-9:20 a.m. in Econ 300 Wednesdays 11:00-12:20 in Econ 300 1
Identification using nonrecursive structure, long-run restrictions and heteroskedasticity 2
General statement of problem of structural interpretation: Can observe in the data: 1t,..., nt errors I make forecasting variables from lagged values. 3
Think of these as resulting from n structural shocks: u 1t shock to technology u 2t shock to price markup u 3t shock to monetary policy t Hu t 4
t Hu t E t t (can observe in the data) Eu t u t D (unknown variance of structural shocks) HDH 5
A. Nonrecursive Orthogonalized VARs Structural model: B 0 y t Bx t u t x t 1, y t1, y t2,...,y tp u t vector of structural shocks Eu t u t D (diagonal) recursive identification assumed B 0 was lower triangular 6
If u t N0,D then log likelihood is Tn/2log2T/2log B 0 1 DB 0 1 1/2 T t1 B 0 y t Bx t D 1 B 0 y t Bx t 7
If B is unrestricted, MLE of B 0 and D are values that maximize T/2log B 0 2 T/2log D T/2traceB 0 D 1 B 0 If model is just identified, estimates will satisfy B 1 1 0 D B 0 8
B. Identification using long-run conditions x t n t log of productivity in quarter t (log GDP minus log of hours worked) log of hours worked in nonag establishments 1948:Q1-1994:Q4 9
y t x t n t I0 VAR (reduced-form) p 4 y t c 1 y t1 2 y t2 p y tp t E t t 10
Structural model: B 0 y t B 1 y t1 B 2 y t2 B p y tp u t Eu t u t I 2 (normalization) 11
Relation between representations: u t B 0 t B 0 1 B 0 1 12
Premultiply structural model, BLy t u t by CL BL 1 : y t C 0 u t C 1 u t1 C 2 u t2 which gives structural MA representation u t u 1t u 2t u 1t u 2t technology shock nontechnology shock 13
Assumption: only technology shocks can have a permanent effect on productivity lim x ts u s 2t 0 14
Notice x ts u 2t x tsx ts1 u 2t x ts1x ts2 u 2t x tx t1 u 2t 15
y t x t x t1 n t n t1 x t x t1 u 2t y 1t u 2t y t C 0 u t C 1 u t1 C 2 u t2 16
y tm u t x ts u 2t C m x tsx ts1 u 2t x ts1x ts2 u 2t is given by the row 1 column 2 element of C 0 C 1 C 2 C s x tx t1 u 2t 17
lim x ts u s 2t 0 requires that the following matrix is lower triangular: C 0 C 1 C 2 C1 18
Goal: find structural disturbances u t that are a linear combination of the VAR innovations, u t B 0 t, such that: (1) Eu t u t I 2 B 0 B 0 I 2 B 1 0 B 1 0 19
(2) y t CLu t (3) C1 is lower triangular 20
Ly t c t t B 1 0 u t Ly t c B 1 0 u t y t L 1 B 1 0 u t y t CLu t C1 1 1 1 B 0 21
C1 1 1 B 0 1 C1C1 1 1 B 0 1 B 0 1 1 1 C1C1 1 1 1 1 Can estimate 1 and from VAR T 1 T t1 t t 1 I 2 1 2 3 4 22
Want: Lower triangular matrix C1 such that C1C1 1 1 1 1 Conclusion: C1 is Cholesky factor of 1 1 1 1 23
To get B 0 we then use fact that C1 1 1 B 0 1 B 0 C1 1 1 1 24
Summary: (1) Estimate VAR by OLS y t x t n t y t ĉ 1y t1 2y t2 py tp t T 1 T t1 t t 25
(2) Find Cholesky factor or lower triangular matrix C such that CC Q Q Q I 2 1 2 p 1 26
(3) Technology shock and transitory shocks for date t are first and second elements of û t B 0 t where B 0 C 1 Q 27
(4) Effect of tech shock and transitory shock at date t on y ts are given by first and second columns, respectively, of y ts u t s B 0 1 28
Estimated structural IRF with 95% CI (Fig 2 in Galí) Figure 2 : US 1.0 technology shock productivity 0.50 transitory shock productivity 0.9 0.8 0.25 0.7 0.00 0.6 0.5-0.25 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12-0.50 0 1 2 3 4 5 6 7 8 9 10 11 12 1.25 gdp 2.00 gdp 1.00 1.75 0.75 0.50 0.25 0.00 1.50 1.25 1.00-0.25 0.75-0.50 0 1 2 3 4 5 6 7 8 9 10 11 12 0.50 0 1 2 3 4 5 6 7 8 9 10 11 12 0.50 hours 2.0 hours 0.25 0.00 1.8 1.6 1.4-0.25 1.2-0.50-0.75 1.0 0.8 0.6 29-1.00 0 1 2 3 4 5 6 7 8 9 10 11 12 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12
Conclusion: technology shock raises productivity and lowers hours worked 30
0. 43 0.06 0. 06 0.42 small negative correlation between VAR resids 31
lim s x ts 1t n ts 1t x ts 2t n ts 2t 1 1 0.89 0. 50 0.87 1. 73 For u 2t to have zero long-run effect on x t, it must be interpreted as something that moves 1t and 2t in the same direction. 32
For u 1t to be uncorrelated with u 2t, it must be interpreted as something that moves 1t and 2t in opposite directions. So if technology shock raises productivity, it must increase 1t and decrease 2t. 33
t B 0 1 u t B 1 0. 59 0.30 0 0. 37 0.53 1t u 1t 2t u 1t 1t u 2t 2t u 2t 0.59 0. 30 0.37 0. 53 34
lim s x ts u 1t n ts u 1t x ts u 2t n ts u 2t 1 1 B 0 1 0.89 0.50 0.87 1.73 0. 71 0 0. 14 1.18 0.59 0.30 0. 37 0.53 35
1t u 1t 2t u 1t 1t u 2t 2t u 2t 0. 59 0.30 0. 37 0.53 A one-unit shock to u 2t is interpreted as causing a 0.30 increase in 1t and 0.53 increase in 2t lim x ts u s 2t 0.890.300. 500.53 0 36
1t u 1t 2t u 1t 1t u 2t 2t u 2t 0.59 0.30 0.37 0.53 A one-unit shock to u 1t is interpreted as causing a 0.59 increase in 1t and 0.37 decrease in 2t n ts u 1t 0.59 n ts 1t 0. 37 n ts 1t 37
1.00 0.75 0.50 0.25 0.00-0.25-0.50-0.75-1.00 1.00 0.75 0.50 0.25 0.00-0.25-0.50-0.75-1.00 1.00 0.75 0.50 0.25 0.00-0.25-0.50-0.75-1.00 Nonorthgonalized response of hours to productivity 0 1 2 3 4 5 6 7 8 9 10 11 12 Nonorthgonalized response of hours to hours 0 1 2 3 4 5 6 7 8 9 10 11 12 Structural response of hours to technology 0 1 2 3 4 5 6 7 8 9 10 11 12 38
Additional comments: (1) Q I 2 1 2 p 1 is estimated poorly, sensitive to p Note: algorithm may be a little more robust if instead use Q 0 1 m for some m 39
Chari, Kehoe and McGrattan (2008) give example of DSGE for which when this kind of procedure is applied does not uncover technology shock. 40
(2) technology shock could be temporary (e.g., delay in adoption of discovered technology) (3) demand shock could be permanent (e.g., lost human capital) 41
C. Identification using heteroskedasticity (Wright, 2012) t daily Nov 3, 2008 to Sep 30, 2011 y 1t 2-year Treasury yield y 2t 10-year Treasury yield y 3t 5-year TIPS break-even (nominal yield minus TIPS yield) y 4t 5-10-year TIPS forward break-even (2 10-year TIPS break-even minus 5-year TIPS break-even) y 5t BAA yield y 6t AAA yield 42
y c t 1 y t1 p y tp t t B 1 0 u t u 1t monetary policy shock want to estimate b 1 (first column of B 1 0
Suppose we believed that: (1) monetary policy shocks have higher variance on particular days Eu 2 1t d 0 11 if t S 0 d 11 if t S Set S is known (Wright uses FOMC dates and dates of monetary policy announcements)
45
(2) A monetary policy shock of given size would have the same effects on these dates as others (3) Variance and effects of other shocks same on these dates as others
Then Eu t u t D e 1 e 1 D if t S if t S e 1 col 1 of I n
t B 1 0 u t n i1 b i u it E t t B 0 1 DB 0 1 b 1 b 1 if t S B 0 1 DB 0 1 if t S
1 1 T 1 T t1 t t t S T 1 T t1 t S 1 0 T 0 T t1 t t t S T 0 T t1 1 0 t S p b 1 b 1 so we can estimate b 1 up to an unknown scale, e.g.: normalize 1
T 1 vech 1 vech 1 L N0, V 1 element of V 1 corresponding to covariance between ij and m given by ( i jm im j (Hamilton, TSA, p. 301).
(1) Test null hypothesis that 0 1 q V 1/T 1 V 0/T 0 1 L q 2 nn 1/2 q vech 1 vech 0 or bootstrap critical value
This statistic is 67.9. asymptotic: P 2 21 46.8 0.001 bootstrap p-value 0. 005 reject H 0 : 0 1 Variance on announcement days different from others so this assumption of framework is correct. 52
(2) Estimate b 1 by minimum chi square: b 1 arg min q V 1/T 1 V 0/T 0 1 q b 1 q q vechb 1 b 1 y ts u 1t sb 1
(3) Test null hypothesis restriction valid: value of objective function asymptotically 2 nn1/2 or bootstrap critical value. This test does not reject assumption that 1 0 b 1 b 1 is consistent with observed data.
Normalization: second element of b 1 0.25 Monetary policy shock lowers 10-year yield by 25 bp. 55
56
No evidence that unconventional monetary policy works through changes in expected inflation or risk premia. 57