**Integrating New-Keynesian Macroeconomics with Term Structure Models for Two-Country Analysis**

1. Combining New-Keynesian Macroeconomics and the Term Structure: A Two-Country ModelLouise LusbyOctober 17,2008

2. Introduction • Recent literature has combined structural New-Keynesian models, featuring aggregate supply, aggregate demand and monetary policy equations, with a no-arbitrage term structure model • Using information from macro variables to explain the dynamics in bond yields • Model: one small open economy (SOE) and one closed economy • Using bond data from two countries gives us additional term structure information • Also, it will allow for a more complete study in the small open economy (SOE); the impact of macroeconomic shocks from both countries on the SOE yield curve can be studied

3. Literature • Term structure literature: latent factors drive the dynamics of the term structure of interest rates • What are the economic forces behind these factors? • VAR models: can identify the sources of the shocks to the selected yields, but they cannot tell how the entire yield curve will respond to those shocks

4. Literature • Ang and Piazzesi (2003) • Bakaert, Cho, and Moreno (2006)

5. Summary Statistics of US Bond Yields(Jan 1990 – Dec 2006) Central MomentsAutocorrelations

6. Summary Statistics of NZ Bond Yields(Jan 1990 – Dec 2006) Central MomentsAutocorrelations

7. The IS Equation • A standard intertemporal IS equation is usually derived from the FOC’s for a representative agent as in the original Lucas (1978) economy. • Problems include: -Pinning down the risk aversion parameter -Matching persistence of output • Therefore, derive an IS equation from utility maximizing framework with external habit formation similar to Fuhrer (2000).

8. The IS Equation • The representative agent maximizes: • Ct is the composite index of consumption • Ft is an aggregate demand shifting factor • ψis the time discount factor • σ is the inverse of the intertemporal elasticity of consumption

9. The IS Equation • Let Ft = Ht Gt • Ht is an external habit level • Gt is an exogenous aggregate demand shock • Following Fuhrer (2000), assume: • η measures the degree of habit dependence on the past consumption level

10. The IS Equation • The Euler equation for the interest rate yields a Fuhrer-type IS equation: • yt is detrended log output • it is the short term interest rate •θ = 1/(σ + η) and μ = σθ • θ measures response of detrended output to the real interest rate • εtIS is an iid demand shock, with homoskedastic variance σ2IS

11. The AS Equation(following Calvo,1983) • Πt is inflation • ytn is the natural rate of output • yt – ytnis the output gap • εtIS is an iid supply shock, with homoskedastic variance σ2AS

12. The Natural Rate of Output • Most studies use an exogenously detrended output variable to serve as the output gap measure in the AS equation. • I will follow Bekaert, Cho, and Moreno (2006) and use an output gap measure that is endogenous and filtered through macro and term structure information. • The natural rate will follow an AR(1) process:

13. The Monetary Policy Rule • Following Clarida, Gali, and Gertler (1999), the monetary authority sets the short-term interest rate according to a forward-looking Taylor rule: Πt*: time-varying inflation target ῑt: desired level of the nominal interest rate that would prevail when EtΠt+1 = Πt* and yt = ytn β : measures the long-run response of the interest rate to expected inflation

14. The Monetary Policy Rule • To capture the tendency of central banks smoothing interest rates: • ρ is the smoothing parameter • εtMPis an iid monetary policy shock • The resulting monetary policy rule is:

15. Inflation Target • I will specify a stochastic process for the inflation target. Define: which can be written as:

16. Inflation Target • I assume that the monetary authority anchors its inflation target around πtLR but smoothes target changes: • Combining equations, we get: where:

17. The Small Open Economy • For the SOE I will follow Svensson (2000) for the aggregate demand equation: • where qt is the (log) real exchange rate • where st is the log nominal exchange rate

18. The Small Open Economy • The AS equation is also of the type used by Svensson (2000): • The timing on exchange rate changes reflects an assumption of instant pass-through

19. The Monetary Policy Rule • Following Svensson (2000): • Formulation assumes closed economy shocks are transmitted to the SOE interest rate through closed economy's monetary policy

20. The Real Exchange Rate • Empirical evidence against UIRP • Therefore, a time-varying risk premium separates expected exchange rate changes from the interest differential • In new open economy macroeconomic models, domestic and foreign macro-variables enter exchange rate equation in differences:

21. The Full Model • 11 variable system • The model can be expressed in matrix form: where: The Rational Expectations (RE) equilibrium can be written as a first-order VAR:

22. Term Structure • I am able to express the short-term interest rate in a country j as: where:

23. The Stochastic Discount Factor • The specification is the standard affine term structure setting. • The pricing kernel process Mt+1 prices all securities such that: where Rt+1 is the total gross return

24. The Term Structure • For an n-period bond: where ptn is the price of an n-period zero- coupon bond at time t

25. The Term Structure • In affine models, the log of the pricing kernel is modeled using a conditionally linear process: where: • is the market price of risk associated with the source of uncertainty, ,in the economy

26. Bond Prices • The SDF in a country prices all zero coupon bonds in that country such that: • We are within the affine class of term structure models because bond prices are exponential affine functions of the state variables.

27. Bond Prices • Bond prices for a country are given by: where the coefficients follow the difference equations:

28. Bond Prices • The continuously compounded yield ytn on an n-period zero coupon bond is given by: • We can see that yields are affine functions of the state variables

29. The Term Structure • I will also follow the standard dynamic arbitrage-free term structure literature and define: • If markets are complete, various papers (Bakaert (1996), Santa-Clara (2002)) demonstrated that this equilibrium condition must hold.

30. Data • Monthly macro, yield, and exchange rate data from New Zealand and the US. • New Zealand has inflation targeting Central Bank • NZ was first country to adopt a formal inflation targeting regime back in 1990. • Sample period: 1990-2008 • Term-structure data at the one, three, five, and ten year maturities.

31. Estimation • Macro parameters • Market prices of risk • Impulse response • Variance decomposition