1. Simulation Methods (cont.) Su, chapters 8-9

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. Reduced Form Equations Y = 50 + 0.2273Y -1 - 0.6818R + 2.2727G C = 44 + 0.1273Y -1 - 0.3818R + 1.2727G I = 6 + 0.1Y -1 - 0.3R T = 10 + 0.0455Y -1 - 0.1364R + 0.4545G

9. 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 )

10. 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

11. Spread Sheet Set-up - Example

12. 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

13. 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!

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

15. 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

16. Time Saving Hint: C Write a formula for (1-  3 )  1 in row 3 Use this and a’s from row 4

17. Multipliers In a dynamic model, can distinguish between two types of multipliers: –Short-term or Impact multipliers –Long-Term Multipliers

18. 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

19. Spreadsheet - Time Paths Put Time and variables in columns Use a’s in formulas to calculate Y,C,I,T

20. Time

21. 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

22. Time Path of Y t - Baseline

23. Additional Policy Simulations Once-for-All Change: G=21 in t+1 only Sustained Change: G=21 in t+1 and all subsequent periods

24. Time Path of Y t - Case 2 & 3

25. 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?

26. 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?