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?