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 Four equation econometric model. –Parameters come from empirical estimates

Model Y  C + I + G C =  1 +  1 (Y - T) I =  2 +  2 Y -1 +  2 R T =  3 +  3 Y Exogenous: Endogenous: Parameters

Structural Model Y  C + I + G C =  1 +  1 (Y - T) I =  2 +  2 Y -1 +  2 R T =  3 +  3 Y Exogenous: G, R Endogenous: Y, C, I, T Parameters:  1  2  3  1  2  3  2

Parameterized Model Y  C + I + G C = 16 +  (Y - T) I = Y -1 -  R T = Y 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 = a 10 + a 11 Y -1 + a 12 R + a 13 G C = a 20 + a 21 Y -1 + a 22 R + a 23 G I = a 30 + a 31 Y -1 + a 32 R + a 33 G T = a 40 + a 41 Y -1 + a 42 R + a 43 G

Reduced Form Equations Y = Y R G C = Y R G I = Y R T = Y R G

Reduced Form Parameters Y The reduced form parameters are functions of the structural parameters Can be solved to get: a 10 =(  1 +  2 -  1  3 ) / (1-  1 +  1  3 ) a 11 =(  2 ) / (1-  1 +  1  3 ) a 12 = (  2 ) / (1-  1 +  1  3 ) a 13 = 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

Time Saving Hint: Y Use Z1 = (1-  1 +  1  3 ), then a 10 =(  1 +  2 -  1  3 ) / Z1 a 11 =(  2 ) / Z1 a 12 = (  2 ) / Z1 a 13 = 1 / Z1 Saves coding steps

Reduced Form Parameters T Want to find these next. Substitute T =  3 +  3 ( a 10 +a 11 Y -1 +a 12 R+a 13 G) a 40 =  3 +  3 a 10 a 41 =  3 a 11 a 42 =  3 a 12 a 43 =  3 a 13 Can use a’s from row 4!

Reduced Form Parameters I These are easy a 30 =  2 a 31 =  2 a 32 =  2 a 33 = 

Reduced Form Parameters C Substitute C =  1 +  1 (Y-T ) C =  1 +  1 [ a 10 +a 11 Y -1 +a 12 R+a 13 G -  3 -  3 ( a 10 +a 11 Y -1 +a 12 R+a 13 G)] a 20 =  1 -  3  3 + (1-  3 )  1 a 10 a 21 = (1-  3 )  1 a 11 a 22 = (1-  3 )  1 a 12 a 23 = (1-  3 )  1 a 13

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

Time

Reduced Form Equation: Y Y = a 10 + a 11 Y -1 + a 12 R + a 13 G $B$4 + $D$4*D9 + $F$4*C10 + $H$4*B10 Use absolute cell references for a’s

Time Path of Y t - Baseline

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 Y t - Case 2 & 3

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?