simulation techniques martin ellison university of warwick and cepr bank of england, december 2005

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Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

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Page 1: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Simulation techniques

Martin Ellison

University of Warwick and CEPR

Bank of England, December 2005

Page 2: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Baseline DSGE model

111*

211*

22*12

*11

*21

1*22

*12

*111

1*21

1*22

*12

*111

*21

1*22

)(

)()(

t

tt

tt

vRPPPP

wPPPPPPPPw

wPPy

Recursive structure makes model easy to simulate

Page 3: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Numerical simulations

Stylised facts

Impulse response functions

Forecast error variance decomposition

Page 4: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Stylised facts

Variances

Covariances/correlations

Autocovariances/autocorrelations

Cross-correlations at leads and lags

Page 5: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Recursive simulation

1. Start from steady-state value w0 = 0

2. Draw shocks {vt} from normal distribution

3. Simulate {wt} from {vt} recursively using

111*

211*

22*12

*11

*21

1*22

*12

*111

1*21

1*22

*12

*111

)(

)()(

t

tt

vRPPPP

wPPPPPPPPw

Page 6: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Recursive simulation

4. Calculate {yt} from {wt} using

5. Calculate desired stylised facts, ignoring first few observations

tt wPPy *21

1*22

Page 7: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Variances

Standard deviation

Interest rate 0.46

Output gap 1.39

Inflation 0.46

Page 8: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Correlations

Interest rate

Output gap

Inflation

Interest rate 1 -1 -1

Output gap -1 1 1

Inflation -1 1 1

Page 9: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Autocorrelations

t,t-1 t,t-2 t,t-3 t,t-4

Interest rate 0.50 0.25 0.12 0.06

Output gap 0.50 0.25 0.12 0.06

Inflation 0.50 0.25 0.12 0.06

Page 10: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Cross-correlations

Correlation with output gap at time t

t-2 t-1 t t+1 t+2

Output gap 0.25 0.50 1 0.50 0.25

Inflation 0.25 0.50 1 0.50 0.25

Interest rate -0.25 -0.50 -1 -0.50 -0.25

Page 11: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Impulse response functions

What is effect of 1 standard deviation shock in any element of vt on variables wt and yt?

1. Start from steady-state value w0 = 0

2. Define shock of interest

0000

0001

0000

}{ tv

Page 12: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Impulse response functions

3. Simulate {wt} from {vt} recursively using

111*

211*

22*12

*11

*21

1*22

*12

*111

1*21

1*22

*12

*111

)(

)()(

t

tt

vRPPPP

wPPPPPPPPw

4. Calculate impulse response {yt} from {wt} using

tt wPPy *21

1*22

Page 13: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Response to vt shock

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8 9 101112

t

interest rate

output gap

inflation

Page 14: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Forecast error variancedecomposition (FEVD)

Imagine you make a forecast for the output gap for next h periods

Because of shocks, you will make forecast errors

What proportion of errors are due to each shock at different horizons?

FEVD is a simple transform of impulse response functions

Page 15: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD calculation

Define impulse response function of output gap to each shocks v1 and v2

28

27

26

25

24

23

22

21

18

17

16

15

14

13

12

11

response to v1

response to v2

response at horizons 1 to 8

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8

t

Page 16: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Shock

Impulse response at horizon 1

Contribution to variance at horizon 1

FEVD at horizon h = 1

At horizon h = 1, two sources of forecast errors

1tv

11 2

1

2tv

2211 1

2221 2v

Page 17: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD at horizon h = 1

Contribution of v1

2221

2211

2211

21

1

)()(

)(

vv

v

Page 18: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Shock

Impulse response at horizon 2

Contribution to variance at horizon 2

FEVD at horizon h = 2

At horizon h = 2, four sources of forecast errors

1tv

12 2

2

2tv

2212 1

2222 2v

11tv 2

1tv

11 2

1

2211 1

2221 2v

Page 19: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD at horizon h = 2

Contribution of v1

2222

2221

2212

2211

2212

2211

2211

11

)()()()(

)()(

vvvv

vv

Page 20: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD at horizon h

Contribution of v1

h

ivi

h

ivi

h

ivi

1

222

1

221

1

221

21

1

)()(

)(

At horizon h, 2h sources of forecast errors

Page 21: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD for output gap

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

h

interest rateshock

cost-pushshock

Page 22: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD for inflation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

h

interest rateshock

cost-pushshock

Page 23: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

FEVD for interest rates

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

h

interest rateshock

cost-pushshock

Page 24: Simulation techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

Next steps

Models with multiple shocks

Taylor rules

Optimal Taylor rules