Simulation Methods (cont.)
Su, chapters 8-9
Numerical Simulation II
• Simulation in Chapter 8, section IV of Su
• Taken from “Forecasting and Analysis with
an Econometric Model,” Daniel B. Suits,
American Economic Review, March 1962,
pp. 104-132
• Four equation econometric model.
– Parameters come from empirical estimates
Model
YC+I+G
C = a1 + b1(Y - T)
I = a2 + b2Y-1 + g2R
T = a3 + b3Y
• Exogenous:
• Endogenous:
• Parameters
Structural Model
YC+I+G
C = a1 + b1(Y - T)
I = a2 + b2Y-1 + g2R
T = a3 + b3Y
• Exogenous: G, R
• Endogenous: Y, C, I, T
• Parameters: a1 a2 a3 b1 b2 b3 g2
Parameterized Model
YC+I+G
C = 16 + 0.7(Y - T)
I = 6 + 0.1Y-1 - 0.3R
T = 0.0 + 0.2Y
Obtained by statistical techniques - data
were obtained and these parameters were
estimated
Reduced Form Equations
• The “solution” to this model is called reduced
form equations
• Shown in equations (8.4a)-(8.4d)
• The numbers are reduced form parameters
• Note that an explicit reduced form equation for Y
has been solved for
• First-order linear difference equations
• Endogenous on RHS, Exogenous on LHS
Reduced Form Equations - General Form
Y
C
I
T
=
=
=
=
a10
a20
a30
a40
+
+
+
+
a11Y-1
a21Y-1
a31Y-1
a41Y-1
+
+
+
+
a12R
a22R
a32R
a42R
+
+
+
+
a13G
a23G
a33G
a43G
Reduced Form Equations
Y
C
I
T
= 50 + 0.2273Y-1
= 44 + 0.1273Y-1
= 6 +
0.1Y-1
= 10 + 0.0455Y-1
- 0.6818R + 2.2727G
- 0.3818R + 1.2727G
0.3R
- 0.1364R + 0.4545G
Reduced Form Parameters Y
• The reduced form parameters are functions of the
structural parameters
• Can be solved to get:
a10=
(a1+ a2- b1 a3) / (1- b1+ b1 b3 )
a11=
(b2) / (1- b1+ b1 b3 )
a12=
(g2) / (1- b1+ b1 b3 )
a13=
1 / (1- b1+ b1 b3 )
Spread Sheet Set-up
• Top 7 rows will be used for parameter
calculations
• Top two rows: Structural Parameters
• Row three: Combinations
• Rows 4-7: Reduced Form parameters
Spread Sheet Set-up - Example
alpha1= 16 alpha2= 6 alpha3= 0 gamma3= -0.3
beta1= 0.7 beta2= 0.1 beta3= 0.2
Z1=
a10=
a11=
a12=
a13=
a20=
a21=
a22=
a23=
a30=
a31=
a32=
a33=
a40=
a41=
a42=
a43=
Time Saving Hint: Y
• Use Z1 = (1- b1+ b1 b3 ), then
a10=
a11=
a12=
a13=
(a1+ a2- b1 a3) / Z1
(b2) / Z1
(g2) / Z1
1 / Z1
• Saves coding steps
Reduced Form Parameters T
• Want to find these next. Substitute
• T = a3+ b3(a10+a11Y-1+a12R+a13G)
a40=
a3+ b3 a10
a41=
b3a11
a42=
b3a12
a43=
b3a13
Can use a’s from row 4!
Reduced Form Parameters I
• These are easy
a30=
a31=
a32=
a33=
a2
b2
g2
0
Reduced Form Parameters C
• Substitute
• C = a1+ b1(Y-T)
• C = a1+ b1[a10+a11Y-1+a12R+a13G a3 - b3(a10+a11Y-1+a12R+a13G)]
a20=
a21=
a22=
a23=
a1- b3 a3 + (1- b3) b1 a10
(1- b3) b1 a11
(1- b3) b1 a12
(1- b3) b1 a13
Time Saving Hint: C
• Write a formula for (1- b3) b1 in row 3
• Use this and a’s from row 4
Multipliers
• In a dynamic model, can distinguish
between two types of multipliers:
– Short-term or Impact multipliers
– Long-Term Multipliers
Baseline Solution
•
•
•
•
“Most likely and reasonable time path”
A basis for comparison
In this case, Y-1 = 100 G=20 R=10
In this case, simply means no change in
fiscal policy
Spreadsheet - Time Paths
• Put Time and variables in columns
• Use a’s in formulas to calculate Y,C,I,T
alpha1=
beta1=
Z1=
a10=
a20=
a30=
a40=
Time
0
1
16
alpha2=
6 alpha3=
0 gamma2=
-0.3
0.7
beta2=
0.1 beta3=
0.2
0.44 (1-beta3)beta1
0.56
50
a11=
0.2273
a12= -0.6818
a13=
2.2727
44
a21=
0.1273
a22= -0.3818
a23=
1.2727
6
a31=
0.1
a32=
-0.3
a33=
0
10
a41= 0.045455
a42= -0.13636
a43= 0.454545
G
R
Y
C
I
T
C+I+G
20
10 100.0000
20
10 111.3636 78.3636 13.0000
22.2727 111.3636
Time
Table 8.1 Spreadsheet
t-1
0
t
1
t+1
2
t+2
3
t+3
4
Reduced Form Equation: Y
• Y = a10 + a11Y-1 + a12R + a13G
• $B$4 + $D$4*D9 + $F$4*C10 + $H$4*B10
• Use absolute cell references for a’s
Time Path of Yt - Baseline
120
115
110
105
100
1
2
3
4
5
6
7
8
9
10 11 12
Additional Policy Simulations
• Once-for-All Change: G=21 in t+1 only
• Sustained Change: G=21 in t+1 and all
subsequent periods
Time Path of Yt - Case 2 & 3
120
115
Series1
110
Series2
Series3
105
100
1
2
3
4
5
6
7
Short and Long Run Multipliers
•
•
•
•
•
What is the Short-Run multiplier on G in
What is the Short-Run multiplier on G in
What is the Long-Run multiplier on G in
What is the Long-Run multiplier on G in
Why the difference?
2?
3?
2?
3?
Summary: Chapter 8 Simulations
• What have we learned about
macroeconomic models?
– Relationship between structural parameters and
reduced form parameters
– How to perform “policy simulations”
• Relationship to Forecasting?