simulation methods (cont.) su, chapters 8-9. numerical simulation ii simulation in chapter 8,...
DESCRIPTION
Model Y C + I + G C = 1 + 1 (Y - T) I = 2 + 2 Y -1 + 2 R T = 3 + 3 Y Exogenous: Endogenous: ParametersTRANSCRIPT
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 = 1 +1(Y - T)
I = 2 + 2Y-1 +2R
T = 3 + 3Y• Exogenous:• Endogenous:• Parameters
Structural Model
Y C + I + G C = 1 +1(Y - T)
I = 2 + 2Y-1 +2R
T = 3 + 3Y• Exogenous: G, R• Endogenous: Y, C, I, T• Parameters: 12 31 2 3 2
Parameterized Model
Y C + I + G C = 16 +(Y - T) I = 6 + 0.1Y-1 -R 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 = a10 + a11Y-1 + a12R + a13GC = a20 + a21Y-1 + a22R + a23GI = a30 + a31Y-1 + a32R + a33GT = a40 + a41Y-1 + a42R + a43G
Reduced Form Equations
Y = 50 + 0.2273Y-1 - 0.6818R + 2.2727GC = 44 + 0.1273Y-1 - 0.3818R + 1.2727GI = 6 + 0.1Y-1 - 0.3RT = 10 + 0.0455Y-1 - 0.1364R + 0.4545G
Reduced Form Parameters Y
• The reduced form parameters are functions of the structural parameters
• Can be solved to get:a10= (1+ 2- 1 3) / (1- 1+ 1 3 )
a11= (2) / (1- 1+ 1 3 )
a12= (2) / (1- 1+ 1 3 )
a13= 1 / (1- 1+ 1 3 )
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= -0.3beta1= 0.7 beta2= 0.1 beta3= 0.2Z1=a10= a11= a12= a13=a20= a21= a22= a23=a30= a31= a32= a33=a40= a41= a42= a43=
Time Saving Hint: Y
• Use Z1 = (1- 1+ 1 3 ), thena10= (1+ 2- 1 3) / Z1
a11= (2) / Z1
a12= (2) / Z1
a13= 1 / Z1
• Saves coding steps
Reduced Form Parameters T
• Want to find these next. Substitute• T = 3+ 3(a10+a11Y-1+a12R+a13G)
a40= 3+ 3 a10
a41= 3a11
a42= 3a12
a43= 3a13
Can use a’s from row 4!
Reduced Form Parameters I
• These are easya30= 2
a31= 2
a32= 2
a33=
Reduced Form Parameters C
• Substitute• C = 1+ 1(Y-T)• C = 1+ 1[a10+a11Y-1+a12R+a13G -
3 - 3(a10+a11Y-1+a12R+a13G)]a20= 1- 3 3 + (1- 3) 1 a10
a21= (1- 3) 1 a11
a22= (1- 3) 1 a12
a23= (1- 3) 1 a13
Time Saving Hint: C
• Write a formula for (1- 3) 1 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= 16 alpha2= 6 alpha3= 0 gamma2= -0.3beta1= 0.7 beta2= 0.1 beta3= 0.2
Z1= 0.44 (1-beta3)beta1 0.56a10= 50 a11= 0.2273 a12= -0.6818 a13= 2.2727a20= 44 a21= 0.1273 a22= -0.3818 a23= 1.2727a30= 6 a31= 0.1 a32= -0.3 a33= 0a40= 10 a41= 0.045455 a42= -0.13636 a43= 0.454545
Time G R Y C I T C+I+G0 20 10 100.00001 20 10 111.3636 78.3636 13.0000 22.2727 111.3636
Time
Table 8.1 Spreadsheett-1 0t 1t+1 2t+2 3t+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
100
105
110
115
120
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
100
105
110
115
120
1 2 3 4 5 6 7
Series1
Series2
Series3
Short and Long Run Multipliers
• What is the Short-Run multiplier on G in 2?• What is the Short-Run multiplier on G in 3?• What is the Long-Run multiplier on G in 2?• What is the Long-Run multiplier on G in 3?• Why the difference?
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?