The changing transmission of uncertainty shocks in the US: an empirical analysis (Technical Appendix)

August 5, 2015

Abstract Technical Appendix

1

Estimation of the FAVAR model

The TVP VAR model is de…ned as

Zt = ct +

P X

tj Zt j

+

j=1

J X

tj

ln

t j

+

1=2 t et

(1)

j=0

where Zt is a matrix of endogenous variables describe below.

(2)

B = vec([c; ; ]) Bt = Bt

1

+

t; V

1

AR ( t ) = QB

t

0

= At 1 Ht At 1

where At is lower triangular. Each non-zero element of At evolves as a random walk

at = at

1

+ gt ; V AR(gt ) = G

(3)

where G is block diagonal as in Primiceri (2005). Following Carriero et al. (0) the volatility process is de…ned as

Ht =

(4)

tS

S = diag(s1 ; ::; sN )

The overall volatility evolves as an AR(1) process

ln

1.1 1.1.1

t

=

+ F ln

t 1

+

t; V

AR( t ) = Q

(5)

Priors and Starting values Factors

We use a principal component estimator to calculate an initial value for the factors Z~t . The initial conditions for the Kalman …lter employed in the Carter and Kohn (2004) step are set as Z0 ~N Z~1 ; I .

2

1.1.2

Factor Loadings and error variances

The initial conditions for the factor loadings are obtained via an OLS estimate of the factor loadings using the …rst 40 observations of the sample period and employing the principal components Z~t . ols

Let

denote the OLS estimate of the factor loadings estimated using the pre-sample data

described above. The prior is set as inverse Wishart Q

;0

s IW Q

;0 ; T0

0 ~N (

ols

where Q

; var(

;0

ols

)): The prior on Q is assumed to be

is assumed to be T0

var(

ols

)

10

4

3:5 and

T0 is the length of the sample used to for calibration. This follows Cogley and Sargent (2005). The prior for the diagonal elements of R is assumed to be IG (R0 ; VR ) where the scale parameter R0 = 0:001 and VR = 5.

1.1.3

VAR Coe¢ cients

The initial conditions for the VAR coe¢ cients B0 are obtained via an OLS estimate of a …xed coe¢ cient VAR using the …rst 40 observations of the sample period. The VAR is estimated using ^ ols and v^ols denote the OLS estimate of the VAR coe¢ cients the principal components Z~t : Let B and the covariance matrix estimated on the pre-sample data described above. The prior for ^ ols ; var(B ^ ols )): The prior on QB is assumed to be inverse Wishart QB;0 s IW QB;0 ; T0 B0 ~N (B where QB;0 is assumed to be T0

^ ols ) var(B

10

4

3:5 and T0 is the length of the sample used

to for calibration.

1.1.4

Elements of the A matrix

The prior for the o¤-diagonal elements At is A0 s N a ^ols ; V a ^ols

where a ^ols are the o¤-diagonal

elements of v^ols , with each row scaled by the corresponding element on the diagonal. V a ^ols is assumed to be diagonal with the elements set equal to 10 times the absolute value of the corresponding element of a ^ols : The prior distribution for the blocks of G is inverse Wishart: Gi;0 s 3

IW (Gi ; Ki ) where i = 1::N

1 indexes the blocks of S: Gi is calibrated using a ^ols . Speci…cally, Gi

is a diagonal matrix with the relevant elements of a ^ols multiplied by 10 3 :

1.1.5

Elements of S and the parameters of the transition equation

The elements of S have an inverse Gamma prior: P (si )~IG(S0;i ; V0 ). The degrees of freedom V0 are set equal to 1. The prior scale parameters are set by estimating the following regression: it

= S0;i

t

+ "t where

t

is the …rst principal component of the stochastic volatilities

it

obtained

using a univariate stochastic volatility model for the residuals of each equation of a VAR estimated via OLS using the endogenous variables Z~t : We set a normal prior for the unconditional mean 0

=

1 F

. This prior is N ( 0 ; Z0 ) where

= 0 and Z0 = 10:The prior for Q is IG (Q0 ; VQ0 ) where Q0 is the average of the variances of

the transition equations of the initial univariate stochastic volatility estimates and VQ0 = 5: The prior for F is N (F0 ; L0 ) where F0 = 0:8 and L0 = 1:

1.1.6

Common Volatility

t

The prior for the initial value of

t

is de…ned as ln

0

N (ln

0 ; I)

where

0

is the initial value of

t:

1.2

MCMC algorithm

The MCMC algorithm is based on drawing from the following conditional posterior distributions ( denotes all other parameters):

1. G( t n ). Given a draw for the factors the variances R and the variance of the shock to the

4

transition equation Q , the following TVP regression applies for the ith Xit :

1=2

Xit =

it Zt

+ Ri "it

=

it 1

+

it

it ; V

AR (

it )

=Q

;i

As this is a linear and Gaussian state space model, the Carter and Kohn (2004) algorithm can be applied to draw from the conditional posterior of

it .

The distribution of the time-

varying loadings conditional on all other parameters is linear and Gaussian: N

T nT ; PT nT

and

tn

t+1; Xit ;

sN

tnt+1;

t+1

; Ptnt+1;

t+1

where t = T

it nXit ;

1; ::1;

s de-

notes a vector that holds all the other VAR parameters: As shown by Carter and Kohn (2004) the simulation proceeds as follows. First we use the Kalman …lter to draw PT nT and then proceed backwards in time using and Ptjt+1;

t+1

= Ptjt

tjt+1;

t+1

=

tjt

1 + Ptjt Pt+1jt

T nT t+1

and tjt

1 Ptjt Pt+1jt Ptjt : Note that in order to deal rotational indeterminancy of

the factors and factor loadings, we …x the …rst K factor loadings where K is the number of factors. In particular, the …rst K

K block of

it

is equal to an identity matrix for all time

periods (see Bernanke et al. (2005)). 2. G(Q ;i n ). Given a draw for

it ,

the conditional posterior for Q

scale matrix and degrees of freedom de…ned as: IW

0 it it

+Q

;i

;0 ; T

is inverse Wishart with + T0 .

3. G(Rn ). The diagonal elements of R have an inverse Gamma conditional posterior:

G(Ri n )~IG ("0it "it + R0 ; T + VR )

4. G (Zn ). Given the parameters of the observation equation ( t ; R) and the transition equa-

5

tion (Bt ;

t ),

equations 9 and 1 constitute a linear Gaussian state space model and the

Carter and Kohn (2004) algorithm can be employed to draw from the conditional posterior distribution of the factors. Carter and Kohn (2004) show that the conditional posterior is de…ned as ZT nXit ; where t = T

1; ::1;

s N ZT nT ; P~T nT

and Zt nZt+1; Xit ;

s N Ztnt+1;Zt+1 ; P~tnt+1;Zt+1

denotes a vector that holds all the other VAR parameters. A

run of the Kalman …lter delivers ZT nT and P~T nT as the …ltered states and its variance at the end of the sample. Then one proceeds backwards in time to obtain Ztnt+1;Zt+1 = 1 Ztjt + P~tjt F~t0 P~t+1jt Zt+1

~t

F~ Ztjt and P~tjt+1;Zt+1 = P~tjt

1 ~ ~ P~tjt F~t0 Pt+1jt Ft Ptjt . Note that F~t

and ~ t denote the coe¢ cients on the lags and the coe¢ cients on pre-determined variables in the transition equation 1 respectively in companion form. 5. G(Bt n ): Given a draw for the factors and variances

t ; QB ,

1 and 2 constitute a VAR with

time-varying parameters and the Carter and Kohn (2004) algorithm can again be applied to draw from the conditional posterior of the VAR coe¢ cients. The distribution of the time-varying VAR coe¢ cients Bt conditional on all other parameters is linear and Gaussian: Bt nZt ; 1; ::1;

s N BT nT ; PT nT and Bt nBt+1; Zt ;

s N Btnt+1;Bt+1 ; Ptnt+1;Bt+1 where t = T

denotes a vector that holds all the other VAR parameters: As shown by Carter and

Kohn (2004) the simulation proceeds as follows. First we use the Kalman …lter to draw BT nT 1 and PT nT and then proceed backwards in time using Btjt+1;Bt+1 = Btjt +Ptjt Pt+1jt Bt+1

Btjt

1 and Ptjt+1;Bt+1 = Ptjt Ptjt Pt+1jt Ptjt : Rejection sampling is used to ensure that the draws satisfy

stability at each point in time. 6. G(QB n ). The draw for QB is standard with conditional distribution IW

0 t t

+ QB;0 ; T + T0 .

7. G(At n ): Given a draw for the VAR parameters and the model can be written as A0t (vt ) = et where vt denotes the VAR residuals. This is a system of linear equations with time-varying 6

coe¢ cients and a known form of heteroscedasticity. The jth equation of this system is given as vjt =

ajt v

columns 1 to j

jt

+ ejt where the subscript j denotes the jth column of v while

1. Note that the variance of ejt is time-varying and given by

varying coe¢ cient follows the process ajt = ajt

1 + gjt

t sj .

j denotes The time-

with the shocks to the jth equation gjt

uncorrelated with those from other equations. In other words the covariance matrix var (g) is assumed to be block diagonal as in Primiceri (2005). With this assumption in place, the Carter and Kohn (2004) algorithm can be applied to draw the time varying coe¢ cients for each equation of this system seperately. 8. G(Sn ). Given a draw for the VAR parameters the model in can be written as A0 (vt ) = et : The jth equation of this system is given by vjt = is time-varying and given by vjt =

ajt v

jt + ejt

where vjt =

t sj : vjt 1=2 t

ajt v

Given a draw for

t

jt

+ ejt where the variance of ejt

this equation can be re-written as

and the variance of ejt is sj . The conditional posterior is

for this variance is inverse Gamma with scale parameter e0jt ejt + S0;j and degrees of freedom V0 + T: 9. G( t n ):Conditional on the VAR parameters, and the parameters of the transition equation, the model has a multivariate non-linear state-space representation. Carlin et al. (1992) show that the conditional distribution of the state variables in a general state-space model can be written as the product of three terms:

~ t nZt ; h

where

~ t nh ~t /f h

~ t+1 nh ~t f h

1

~ t = ln denotes all other parameters and h

t.

~ t; f Z t nh

(6)

In the context of stochastic volatility

models, Jacquier et al. (1994) show that this density is a product of log normal densities for

7

t

and

t+1

and a normal density for Zt :Carlin et al. (1992) derive the general form of the

~ t nh ~ t 1; h ~ t+1 ; mean and variance of the underlying normal density for f h

~ t nh ~t /f h

1

~ t+1 nh ~ t and show that this is given as f h ~ t nh ~ t 1; h ~ t+1 ; f h

where B2t1 = Q

1

(7)

N (B2t b2t ; B2t )

~ t 1F 0Q + F 0 Q 1 F and b2t = h

1

~ t+1 Q 1 F: Note that due to the non+h

linearity of the observation equation of the model an analytical expression for the complete ~ t nZt ; conditional h

is unavailable and a metropolis step is required. Following Jacquier et al.

(1994) we draw from 6 using a date-by-date independence metropolis step using the density in 7 as the candidate generating density. This choice implies that the acceptance probability ~ t; is given by the ratio of the conditional likelihood f Zt nh

at the old and the new draw.

~ = ln To implement the algorithm we begin with an initial estimate of h

t

We set the matrix

~ old equal to the initial volatility estimate. Then at each date the following two steps are h implemented:

~ new using the density 6 where b2t = h ~ new F 0 Q (a) Draw a candidate for the volatility h t t 1 ~ old Q 1 F and B 1 = Q h 2t t+1

1

likelihood of the VAR for observation t and de…ned as j ct +

PP

j=1

+

+ F 0Q 1F

~ new ~ t; ~ old = h ~ new with acceptance probability f (Zt nht ; ) where f Zt nh (b) Update h t t ~ old ; ) f (Zt nh t

e~t = Zt

1

tj Zt j

+

PJ

tj

j=0

ln

t j

and

tj t

0:5

0:5 exp e~t

t

is the

1 0 e~t

where 0

~ t )S At 1 = At 1 exp(h

Repeating these steps for the entire time series delivers a draw of the stochastic volatilties.1 1

In order to take endpoints into account, the algorithm is modi…ed slightly for the initial condition and the last observation. Details of these changes can be found in Jacquier et al. (1994).

8

7. G( ; F n ):We re-write the transition equation in deviations from the mean ~t h

~t =F h

where the elements of the mean vector and

i

+

1

are de…ned as

(8)

t

i

1 Fi

~t : Conditional on a draw for h

the transition equation 8 is a simply a linear regression and the standard normal and

~ = Fh ~ inverse Gamma conditional posteriors apply. Consider h t t ~ =h ~t h t

~ ;h t

1

~t =h

L

1

+

t; V

AR ( t ) = Q and

: The conditional posterior of F is N ( ; L ) where

1

=

L0 1 +

1 ~ 0 ~ h h Q t 1 t

=

L0 1 +

1 ~ 0 ~ h h Q t 1 t

1

L0 1 F0 +

1

1 ~ 0 ~ h h Q t 1 t

1 1

The conditional posterior of Q is inverse Gamma with scale parameter

0 t t

+ Q0 and degrees of

freedom T + VQ0 . Given a draw for F , equation 8 can be expressed as and C = 1

F: The conditional posterior of

Z

Note that

1

1 0 + CC Q

1

Z0 1 +

=

1

can be recovered as

Z0

(1

t

is N ( ; Z ) where

1 0 CC Q

=

~t = C + h

Z0 1

)

9

0

+

1 0 ~ C ht Q

where

~t = h ~t h

~t Fh

1

1.3

A Monte-Carlo experiment

In order to evaluate the MCMC algorithm we conduct a simple Monte Carlo experiment. 400 observations are generated from the following DGP with the number of variables N = 40 and the number of factors k = 2. The …rst 100 observations are discarded to remove the impact of initial conditions. Xit =

Zt =

t Zt 1

Ht =

+

t Fkt

t

ln

S 0t

+ uit ; uit ~N (0; 1)

t

1

B 1 0 C C S = B A @ 0 1 t

t

where

t

=

0:1 + 0:75 0

B = B @

1=2 t et ; et ~N (0; 1)

+ ct +

11;t

21;t

1

+ (0:5) 2 vt 1 0

t 1

12;t

22;t

C C; A

t

B =B @

11;t

21;t

1 C C A

is generated once using vt ~N (0; 1) and …xed for all iterations of the experiment. Following

Gamble and LeSage (1993) we assume that a one time shift de…nes the change in the factor loadings, the VAR coe¢ cients and 0 the non-zero 1 element 0 of At .1During the …rst 100 observations

the VAR coe¢ cients equal matrix 0 t

t

is equal 1 to

B 0:5 0:05 C C; =B @ A 0:05 0:5

1~

t

t

B 0:5 0:05 C C; =B @ A 0:05 0:5

t

B =B @

0:5 C C and A = A 0:5

1. The factor loading

N0(0; 1). During the next 300 observations, the coe¢ cients change to 1

B =B @

1:5 C C, A = 1 and A 1:5 10

t

=

2~

N (0; 1). Note that the factor loadings

are generated once and held …xed over the Monte-Carlo iterations. The data is generated 100 times. For each replication, the MCMC algorithm described above is run using 5000 iterations and the last 1000 draws are used to compute the impulse response to a one standard deviation shock to the volatility

t.

Note that we use the …rst 20 observations to

calibrate priors and starting values. Figure 1 plots the median estimate of the cumulated impulse response of Xit (for i = 1; 2; ::40) at the 4-period horizon across Monte-Carlo replications and compares these with the true underlying values of the response (solid black lines). The …gure shows that in the case of most variables, the Monte-Carlo estimates of the shift in the response matches the change in the response assumed in the DGP.

1.4

DIC Calculation

In practical terms, the DIC can be calculated as: DIC = D + pD . The …rst term is de…ned as D = E ( 2 ln L ( i )) = of all of the parameters

1 M i

P

i

( 2 ln L ( i )) where L ( i ) is the likelihood evaluated at the draws

in the MCMC chain. This term measures goodness of …t. The second

term pD is de…ned as a measure of the number of e¤ective parameters in the model (or model complexity). This is de…ned as pD = E ( 2 ln L ( i )) as pD =

1 M

P

i

( 2 ln L ( i ))

2 ln L

1 M

P

i

i

evaluated using a particle …lter with 2000 particles.

11

( 2 ln L (E( i ))) and can be approximated . The likelihood function of the model is

12

Figure 1: Estimates of the cumulated impulse response at the 4-period horizon. The …gure presents the median (red line ) across the 100 Monte-Carlo replications. The black line represents the true time-varying cumulated response at the 4-period horizon.

1.5

Impulse responses from the benchmark model (full 3-D …gures)

13

14

Figure 2: Three dimensional version of the impulse responses from the benchmark model

1.6

Sensitivity analysis

The top row of …gure 3 in section 1.6 presents the impulse response of key series to a 1 standard deviation uncertainty shock using a version of the benchmark model where the four lags of

t

are

allowed to a¤ect the endogenous variables. As in the benchmark case, the response of GDP the corporate bond spread and stock market returns show a decline. In contrast, the decline in the in‡ation and the short term interest rate response is estimated to be weaker. The second row of …gure 3 shows the impulse response from a version of the benchmark model where the assumption that the volatility has a contemporaneous a¤ect on the endogenous variables is relaxed. In contrast, only the coe¢ cients on lagged values of

t

are allowed to have non-zero

coe¢ cients. The temporal pattern of the estimated impulse responses support the benchmark results– while the response of GDP growth and the …nancial variables declines over time, the response of in‡ation and the short-term rate is fairly stable. The third row of the …gure shows that the responses from a three factor model support the key conclusions reached using the benchmark model. The fourth row of the …gure considers a version of the benchmark model where an alternative prior imposed on the variance of the shock to the transition equations for the time-varying parameters (QB and Q ). In the benchmark model, the prior for these covariance matrices is assumed to be inverse Wishart: P (QB ) ~IW (QOLS

T0

K; T0 )

where T0 = 40 is the length of the training sample and QOLS is the OLS estimate of the coe¢ cient covariance using the training sample. The prior scale matrix is multiplied by the factor K which is set to 3:5 K =1

10

10 4

4

following Cogley and Sargent (2005). In the alternative speci…cation, we set

and thus incorporate a belief of lower time-variation in the VAR coe¢ cients and 15

factor loadings. The bottom panel of …gure 3 shows that while the change in impulse responses is smoother, there is evidence that the estimated response of GDP growth, the corporate bond spread and stock price index declines over time. In contrast, the response of in‡ation and interest rates remains largely constant. Thus, the results from this model with a tighter prior support the benchmark conclusions. The bottom row of the …gure extends the benchmark model by adding stochastic volatility in the observation equation. In particular, the observation equation is modi…ed as follows:observation equation Xit =

t Zt

1=2

+ Rt "it

(9)

where the diagonal elements of Rt follow a stochastic volatility process:

ln rit = ln rit

1=2

1

+ Gi ot

where rit denotes the diagonl elements of Rt : The responses shown in the last row are very similar to the benchmark responses suggesting that the results are robust to this change in the model speci…cation. In summary, the benchmark results and the sensitivity analysis suggests the following main conclusion: There is evidence that the response of real activity and some …nancial indicators to the uncertainty shock has declined over time. In contrast, the response of in‡ation and interest rates to this shock has remained largely stable. We now turn to a DSGE model in order to explore the possible reasons behind the estimated change in the impact of uncertainty shocks.

16

17 Figure 3: Sensitivity Analysis

1.7

Data

18

19

Table 1: List of Data series. GFD refers to Global Financial data. FRED is the St Louis Fed database. LD denotes 100 times the log di¤erence while N denotes no transformation No. Variable Source Code Transformation 1 Industrial Production FRED INDPRO LD 2 Dow Jones Industrial Total returns index GFD _DJITRD LD 3 GDP De‡ator FRED GDPDEF LD 4 ISM Manufacturing: New Orders Index FRED NAPMNOI N 5 ISM Manufacturing: Inventories Index FRED NAPMII N 6 ISM Manufacturing: Supplier Deliveries Index FRED NAPMSDI N 7 All Employees: Total nonfarm FRED PAYEMS LD 8 Business Con…dence Index GFD BCUSAM N 9 Real Imports FRED IMPGSC96 LD 10 Real Exports FRED EXPGSC1 LD 11 Government Spending to GDP ratio BEA see Mumtaz and Surico (2013) LD 12 Net Taxes to GDP ratio BEA seeMumtaz and Surico (2013) LD 13 Real Gross Private Domestic Investment FRED GDPIC96 LD 14 Real Personal Consumption Expenditures FRED PCECC96 LD 15 Real GDP FRED GDPC96 LD 16 Unemployment Rate FRED UNRATE N 17 Average Hours FRED AWHMAN LD 18 Civilian Labour Force FRED CLF16OV LD 19 Civilian Labor Force Participation Rate FRED CIVPART LD 20 Nonfarm Business Sector: Unit Labor Cost FRED ULCNFB LD

20

Table 2: List of Data series. GFD refers to Global Financial data. FRED is the St Louis Fed database. LD denotes 100 times the log di¤erence while N denotes no transformation (continues) No. Variable Source Code Transformation 21 Nonfarm Business Sector: Real Compensation Per Hour FRED COMPRNFB LD 22 M2 Money Stock Fred M2 LD 23 Total Consumer Credit Owned and Securitized, Outstanding FRED TOTALSL LD 24 Producer Price Index GFD WPUSAM LD 25 CPI FRED CPIAUCSL LD 26 Personal Consumption Expenditures: Chain-type Price Index FRED PCECTPI LD 27 3-Month Treasury Bill: Secondary Market Rate FRED TB3MS N 28 10 year Govt Bond Yield minus 3 mth yield GFD IGUSA10D (minus TB3MS) N 29 6-month Treasury bill minus 3 mth yield GFD ITUSA6D (minus TB3MS) N 30 1 year Govt Bond Yield minus 3 mth yield GFD IGUSA1D (minus TB3MS) N 31 5 year Govt Bond Yield minus 3 mth yield GFD IGUSA5D (minus TB3MS) N 32 Reuters/Je¤ries-CRB Total Return Index (w/GFD extension) GFD _CRBTRD LD 33 West Texas Intermediate Oil Price GFD _WTC_D LD 34 BAA Corporate Spread GFD MOCBAAD (minus IGUSA10D) N 35 AAA Corporate Bond Spread GFD MOCAAAD (minus IGUSA10D) N 36 S&P500 Total Return Index GFD _SPXTRD LD 37 NYSE Stock Market Capitalization GFD USNYCAPM LD 38 S&P500 P/E Ratio GFD SYUSAPM N 39 US Canada exchange rate] GFD USDCAD LD

Figure 4: Recursive means of the Gibbs draws calculated at every 100 draws.

1.8

Recursive means

Figure 4 presents the mean of the Gibbs draws for key model parameters calculated every 100 draws. These are fairly stable which provides evidence of convergence of the Gibbs algorithm.

2 2.1

Details on the DSGE model Calibration

The benchmark calibration is listed in table 4. The parametrization of the non-…nancial block of the of the model is based on Fernandez-Villaverde and Rubio-Ramirez (2008), while the calibration of the …nancial block of the model relies on Christiano et al. (2014). In both studies full information Bayesian estimation techniques have been used to decide about the structural parameter and this is what drives our selection. However, for the purposes of the simulation experiments discussed in the 21

main text we have changed the values of

, ,

p,

,

A,

and relative to the numbers reported

by Fernandez-Villaverde and Rubio-Ramirez (2008) (Table 2.1). To be precise, the discount factor is set equal to 0.999 and combined with the in‡ation target investment speci…c technological change

= 1:010, the growth rates of the

= 1:010 and of the neutral technology

A

= 1:0013,

implies that the steady-state value of the annual real rate is 6:40%. The degree of habit persistence is 0:88, this value is higher than the estimates reported by Smets and Wouters (2007) and Justiniano et al. (2010), however, due to log consumption preferences a high degree of habit persistence is needed so demand does not display excess sensitivity to the real interest rate (via the Euler consumption equation). Similar to Smets and Wouters (2007) the inverse Frisch elasticity of labour supply is equal to 1:36 and the investment adjustment is equal to 7:68 suggesting very little response of investment to changes in Tobin’s q. The Calvo parameters imply that prices and wages are reset every 2:22 and 3:33 quarters respectively, while households rely on indexation ( more heavily than …rms ( = 0:40). The elasticities of substitution for …rms and

w

= 0:80)

for households

imply an average markup of around 5% and 30%, respectively. The Taylor rule parameters are = 1:010,

y

= 0:190 and

R

= 0:79. The steady-state probability of defaults is 2:24% and

slightly smaller than the value 3% used in the literature (see Bernanke et al. (1999)). The value of entrepreneur auditing cost is 0:21 and the fraction of survival entrepreneurs is 0:985, again these values are higher than those used in the literature (3% and 0:976 Bernanke et al. (1999)).

2.2

Solution

The model is solved using third-order perturbation methods (see Judd (1998)) as for any order lower than three, uncertainty shocks (our main objects of interest) do not enter into the decision rule as independent components. One di¢ culty of using these higher-order solution techniques is that paths simulated by the approximated policy function often explode. This is because regular 22

perturbation approximations are polynomials that have multiple steady states and could yield unbounded solutions (Kim et al., 2008). This means that the approximation is valid only locally and along the simulation path we may enter into a region where its validity is not preserved anymore. To avoid this problem Kim et al. (2008) suggest to ‘prune’all those terms that have an order that is higher than the approximation order, while Andreasen et al. (2013) show how this logic can be applied to any order. Although there are studies that question the legitimacy of this approach (see Haan and Wind (2010)), it has by now been widely accepted as the only reliable way to get the solution of nth order approximated DSGE model (where n > 1). Finally, we follow Fernández-Villaverde et al. (2011) and generate the responses of model variables to stochastic volatility shocks using generalised impulse responses developed by Koop et al. (1996).

2.3

IRFs

The exact simulation steps to produce the impulse responses reported here are as follows:

1. We draw 5

1040 structural shocks ! j;t from the standard normal distribution (where j =

1; ::; 5 and t = 1; ::; 1040) 2. We simulate the model using the shocks from step 1, we denote the simulated data by yt 3. We simulate the model using again the structural shocks ! j;t from step 1 but now we increase the value of the structural shock of interest in period 1001 by an amount necessary to rise uncertainty by 1 times the standard devition of the uncertainty shock, namely

! ~ j;1001 = ! j;1001 + x 23

(10)

We denote the data obtained from this simulation by y~t 4. Steps 1 to 3 are repeated 1000 times 5. The IRF is calculated as follows 1 X i y~ 1000 i=1 t 1000

IRF =

yti

All calculations have been produced using Dynare 4.4.2 and Matlab 2012b.

24

(11)

Table 3: DSGE Model Variables Description Mnemonic dt Preference Shock ct Consumption Trend Growth Rate of the Economy z;t Growth rate of Investment-Speci…c Technology Growth I;t Growth rate of Neutral Technology A;t Lagrange multiplier t Rt Nominal Interest rate In‡ation t rt Rental Rate of Capital xt Investment ut Capacity Utilization qt Tobin’s Marginal Q ft Variable for Recursive Formulation of Wage Setting d Aggregate Labor Demand lt wt Real Wage wt Optimal Real Wage w; Optimal Wage In‡ation t Optimal Price In‡ation t 1 Variable 1 for Recursive Formulation of Price Setting gt 2 gt Variable 2 for Recursive Formulation of Price Setting ytd Aggregate Output mct Marginal Cost kt Capital p vt Price Dispersion Term w Wage Dispersion Term vt lt Aggregate Labor Bundle Labor Disutility Shock t Ft Firm Pro…ts Aggregate Uncertainty t !t Bankruptcy Cuto¤ Value Rtk Return of Capital nt Net worth

25

Table 4: DSGE Model Parameters Description Mnemonic Value Steady State Growth Rate of Investment-Speci…c Technology 1.010 Steady State Neutral Technology Growth 1.005 A Discount Factor 0.999 h Consumption Habits 0.877 Capital Utilization, Linear Term 0.039 1 Capital Utilization, Quadratic Term 0.001 2 Depreciation Rate 0.015 Capital Adjustment Cost Parameter 7.679 Elasticity of Substitution between Labor Varieties 4.20 Elasticity of Substitution between Goods Varieties 21.00 Labor Disutility Parameter 9.340 Inverse Frisch Elasticity 1.359 Wage Indexation Parameter 0.800 w Price Indexation 0.400 Calvo Probability Prices 0.550 p Calvo Probability wages 0.700 w Capital Share 0.255 Steady State In‡ation 1.010 Interest Smoothing Coe¢ cient Taylor Rule 0.790 R Feedback In‡ation Coe¢ cient Taylor Rule 1.010 Feedback Output Coe¢ cient Taylor Rule 0.190 y Firms Fixed Cost 0.025 Financial Friction Parameters F (!) Steady State Probability of Default 0.006 Fraction of Survival Entrepreneurs 0.985 Entrepreneur Auditing Cost 0.210 E ! Steady State Bankruptcy Cuto¤ Value 0.568 Standard Deviation Entrepreneur’ s Idiosyncratic Productivity Shock 0.214 ! Fraction of Assets Consumed During Exit 0.005 Notes: The parametrization of the model is based on Fernandez-Villaverde and Rubio-Ramirez (2008), while the values of the …nancial friction parameters are those estimated by Christiano et al. (2014). We refer to this version of the model in the text as ‘benchmark’ and for the purposes of the simulation experiments discussed in Section xxx we have changed the values of , , p , , A , and relative to the numbers reported by Fernandez-Villaverde and Rubio-Ramirez (2008) (Table 2.1). 1 is selected to deliver a steady-state (annual) credit spread is equal to 300bps, ensures the steady-state value of labour is equal to 1=3, is selected so the steady-state value of pro…ts is equal to zero, ! and ! have been also selected in order to be consistent with F (!) = 0:0056 and E = 0:210.

26

Table 5: DSGE Model Shock Process Parameters Description Mnemonic Value Autocorrelation Preference Shock 0.951 d Autocorrelation Labor Disutility Shock 0.942 Autocorrelation Uncertainty Shock 0.950 standard deviation preference shock 0.060 d Standard Deviation labor Disutility Shock 0.070 Standard Deviation Investment-Speci…c Technology 0.151 Standard Deviation Neutral Technology 0.070 A Standard Deviation Policy shock 0.003 m Standard Uncertainty Shock 1.000

27

Table 6: DSGE Model Equations Description

Equation

Marginal Utility of Consumption

d t ct

Euler Equation Rental Rate of Capital Return of Capital

t+1 Rt = Et zt+1 t+1 0 rt = (ut+1 )

Investment Equation

1=

1

h ctzt 1

h Et dt+1 (ct+1 zt+1

hct )

1

=

t

t

Rtk t

rt ut +(1 )qt a(ut ) n qt 1 t qt 1 S xxtt zt1

xt+1 zt+1 xt

S0 t zt+1 t+1

+ Et

xt z t xt 1

S0

xt zt xt 1

xt+1 zt+1 xt

o

2

Wages Equation 3

(1 ) w w t+1 zt+1 d t t wt lt + w Et wt t+1 (1+ ) d 1+ ) ft = dt t ( w; ) (l t t (1+ ) w (1+ ) w t+1 zt+1 t + w Et t+1 ft+1 wt 1 1 w w; 1 wt 1 1 = w t t1 + (1 w )( t ) wt zt

Prices Equation 1

gt1 =

Prices Equation 2 Prices Equation 3

gt2 = t gt1 = (

Prices Equation 4

1=

Demand for Capital

ut kt ltd

Marginal Cost

mct =

Wages Equation 1

1

ft =

Wages Equation 2

Rt R

Taylor Rule

(wt )(1

d t mct yt

+

wt zt rt 1

1 1 1

R

Rt 1 R

=

ytd = ct + xt + lt = vtw ltd

Price Dispersion

vtp =

p

Wage Dispersion

vtw =

w

Capital Accumulation

kt = (1

Bank zero pro…t condition

[ (! t ) 8 < Et [1 :

Notes: G (!; distribution.

!)

=1

@ (! t ) @ !t

and

nt = 0:5

log !

! !

@G(! t ) @ !t

,

wt1

1

(

rt ytd 1

! y )1

1

1

1

w t 1

k E G(! t )Rt qt 1 kt 1

+

+ (1

t

zt

+ 1

xt z t xt 1

S

@

(! t+1 )]

t

w; w) ( t

k qt 1 kt E G (! t )] Rt nt 1

1

nt t zt

(! t ) = ! t [1

1

k Rt+1 Rt

+

+

[Rtk (Rt

+

1

(nt

wE )

1

xt

= Rt

(!t+1 ) @ ! t+1

qt

1

1 kt 1

nt

1

1

[ (! t+1 )

E G(! t+1 )]

(!t+1 )

@G(! t+1 ) E @ ! t+1

@

k 1 + E G(! t )Rt t zt

F (! t )] + G (! t ), where

denote the partial derivatives of

28

emt

)

@ ! t+1

Rt

R

p) ( t )

+ (1

t

t

ytd zt 1 1 A

t

(ut )kt zt t

1

2 gt+1

t t+1

(ltd )

t 1

wt 1 wt zt ) kztt

ft+1

t

1 A (u k ) zt t t t 1 vtp

Aggregate Demand Labour Market Clearing

1)

1 p )( t )

+ (1

t

=

t t+1

1

t 1

ytd =

Net-worth accumulation

p Et

(

1 gt+1

t t+1

1

+ 1)gt2

Goods Market Clearing

Optimality loan contract condition

p Et

d t yt

p 1

)

)]

qt 1 kt

1

k Rt+1 Rt

1

+ wE

9 = ;

is the CDF of a normal

(! t ) and G (! t ) with respect to ! t .

=0

Table 7: DSGE Model Exogenous States Equations Description Preference shock Labour disutility shock Policy Shock TFP Shock IST Shock Uncertainty Shock

Equation log (dt ) = (1 d ) log(d) + d log (dt 1 ) + t log( ) + log t 1 + log ( t ) = 1 log(mt ) = t m ! m t log (At ) = log( A ) + t A ! A t log ( t ) = log( ) + t ! A t log ( t ) = log ( t 1 ) + ! t

29

d d!t t

!t

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