The Delta Method ECON Tori Kreinbrink & Shawn Enriques December 7, 2017

Transcription

1 The Delta Method ECON 5350 Tori Kreinbrink & Shawn Enriques December 7, 2017

2 Table of contents 1. Background 2. Delta Method Explained 3. Example 4. Matlab 1

3 Background

4 Review Concepts Taylor Series Approximation Central Limit Theorem & Asymptotic Normality 2

5 Review We often encounter nonlinear functions in econometrics To handle these, we can take Taylor expansion of the function around the true parameter This linear approximation can be used for further analysis 3

6 Taylor Series Approximation The nonlinear function f(x) needs to be approximated. 4

7 Taylor Series Approximation Taylor s theorem says that there exists a c such that f f(b) f(a) (c) =. b a 5

8 Central Limit Theorem & Asymptotic Normality As n, the distribution becomes a normal 6

9 Background What is the delta method used for? The delta method is often used to estimate standard errors. n( ˆβ β 0 ) d N(0, V) ( ) n(f( ˆβ) f(β 0 )) d df 2 N(0, ˆβ=β 0 V) 7

10 Delta Method Explained

11 The Linear Case Let X 1, X 2,...X n N(10, σ 2 = 2). Or in short, we have X i N(10, 2). We can take the linear transformation of this Y i = 4X i + 3 8

12 The Linear Case For real scalar constants a and b we can show that E(a) = a E(aX + b) = ae(x) + b var(a) = 0 var(ax + b) = a 2 var(x) Thus we find, Y N([4(10) + 3 = 43], [(4 2 )(2)]) Y N(43, 32) 9

13 The Nonlinear Case Any differentiable function over a narrow enough region appears approximately linear. The approximating line is the tangent line to the curve, and its slope is the derivative of the function. 10

14 When do we use the Delta Method? We can derive an approximation to the same tangent line by using a Taylor series expansion of f( ˆβ) aroun = β 0 : γ = f( ˆβ) f(β 0 ) + df ˆβ=β ( ˆβ β 0 0 ) = f(β 0 ) + df ˆβ=β ˆβ df 0 ˆβ=β β 0 0. Rearranging this and taking the limits we get: n(f( ˆβ) f(β 0 )) = a df ˆβ=β0 n( ˆβ β 0 ) 11

15 When do we use the Delta Method? We can derive an approximation to the same tangent line by using a Taylor series expansion of f( ˆβ) aroun = β 0 : E(γ) df ˆβ=β β f(β 0 ) df ˆβ=β β 0 0 = f(β 0 ) var(γ) var(f( ( ˆβ)) = f( ˆβ) f(β 0 ) ) 2 = ( df ˆβ=β ( ˆβ β 0 0 )) 2 ( ) df 2 = ˆβ=β0 ( ˆβ β 0 ) 2 ( ) df 2 = ˆβ=β var( ˆβ) 0 12

16 Example

17 Example Government spending multiplier The amount by which equilibrium real GDP changes as a result of a $1 change in autonomous consumption, investment spending, government purchases or net exports. 1, where MPC is the Marginal Propensity to 1 MPC Consume 13

18 Example Government spending multiplier Let s set MPC = ˆβ. We know that that the government spending multiplier is a nonlinear function of the MPC. Let s call this f( ˆβ). 14

19 Government spending multiplier If we know ˆβ N(β 0, σ 2 ) We can do a first-order Taylor series expansion f( ˆβ) f(β 0 ) + df ˆβ=β0 ( ˆβ β 0 ) = f(β 0 ) + df ˆβ=β ˆβ df 0 ˆβ=β β 0 0. Rearranging this and taking the limits we get: n(f( ˆβ) f(β 0 )) = a df ˆβ=β0 n( ˆβ β 0 ) 15

20 Government spending multiplier From the above we can see E(f( ˆβ)) = f(β 0 ) ( ) df 2 var(f( ˆβ)) = var( ˆβ) ˆβ=β 0 Plugging back in... MPC N(MPC0, σ 2 ) ( ) f( MPC) df 2 N(f(MPC 0 ), σ 2 d MPC MPC=MPC0 ) 16

21 Matlab

22 Matlab- Example Model Set-up Suppose we want to find the Spending Multiplier for GDP and do a significance test on its value. The Marginal Propensity to Consume, MPC, is defined as the rate of change of spending from a change in income. MPC = C Y And the Spending Multiplier is a simple transformation of the MPC 1 MULT = 1 MPC 17

23 Matlab- The Problem In order to do the hypothesis test, we need to generate the t-statistic. It is given by t = MULT µ s e ( MULT) s e ( MULT) is NOT KNOWN, but the Delta Method can estimate it. 18

24 Matlab- Example Set-up Because the spending multiplier is not a linear function, it is difficult to estimate the variance. Here s a good example of when to use the Delta Method to estimate it. The MPC along with its variance can be estimated using the this model C t = β 1 + β 2 Y t + ϵ t Using data from FRED for per capita income and personal consumption we can estimate β 2, which is the MPC

25 Matlab- Income and Spending Data Here is a scatter plot of consumption and income. Here, the MPC is the slope of the regression line, β 2. With a variance of σ 2 (X X) 1. 20

26 Matlab- The Delta Method to Estimate Variance After completing the OLS estimate for the MPC (β 2 ), we can use the delta method to estimate the variance of the Spending Multiplier, MULT. Recall that from the Delta method, s e ( MULT) = f (x) 2 V x where, f(mpc) = 1 1 MPC, df dx = 1 (1 MPC) 2 21

27 Matlab- The Solution The hypothesis test can now be completed. Let H o : MULT = 5 and H a : MULT 5 t = MULT µ s e ( MULT) t = MULT µ 2 df Var MPC dmult Lets do some tweaking of the parameters in the Matlab Code. 22