simulation techniques martin ellison university of warwick and cepr bank of england, december 2005
TRANSCRIPT
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
Numerical simulations
Stylised facts
Impulse response functions
Forecast error variance decomposition
Stylised facts
Variances
Covariances/correlations
Autocovariances/autocorrelations
Cross-correlations at leads and lags
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
Recursive simulation
4. Calculate {yt} from {wt} using
5. Calculate desired stylised facts, ignoring first few observations
tt wPPy *21
1*22
Variances
Standard deviation
Interest rate 0.46
Output gap 1.39
Inflation 0.46
Correlations
Interest rate
Output gap
Inflation
Interest rate 1 -1 -1
Output gap -1 1 1
Inflation -1 1 1
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
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
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
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
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
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
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
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
FEVD at horizon h = 1
Contribution of v1
2221
2211
2211
21
1
)()(
)(
vv
v
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
FEVD at horizon h = 2
Contribution of v1
2222
2221
2212
2211
2212
2211
2211
11
)()()()(
)()(
vvvv
vv
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
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
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
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
Next steps
Models with multiple shocks
Taylor rules
Optimal Taylor rules