Hi,

I can solve the model in dynare but I need your help for the following problem: How does one derive a log-linearized expression for a forward-looking variable around the steady-state? For example, the price dividend ratio can be defined as

\frac{P_t}{D_t} = E_t [M_{t,t+1}\frac{D_{t+1}}{D_t}(1+
\frac{P_{t+1}}{D_{t+1}})]

The goal is to derive the relation between \frac{P_{t+1}}{D_{t+}} and shocks at t+1, assuming that at t the system is in steady state so \frac{P_t}{D_t} is the steady state value C.

If we ignore the expectation and log-linearize it, I get

\log C = m_{t,t+1} + \Delta D_{t+1} + \log(1+\frac{P_{t+1}}{D_{t+1}})

and clearly one can solve \frac{P_{t+1}}{D_{t+1}} analytically (assuming that m and \Delta D are known functions of shocks).

But, isn’t ignoring expectation imposing that the equality hold **state by state**, instead of on average?

Please advise, thanks!

You linearize within the expectations (Leibniz Rule), so it stays there.

Thanks for your reply. I think I understand that in dynare I should put it that way.

But I was curious if my goal was to understand how \frac{P}{D} depend on other state variables (analytical expression), can we rely on this log-linearization? As you said, it still has a Leibniz rule so the integral is still there.

How to get a linear relationship between \frac{P}{D} and other state variables?

Do we need to assume a linear relationship between PD and state variables as in the LRR paper? Thanks!

I don’t think it is correct to think of it as “ignoring the expectation”. Like Johannes says, you are linearizing *within* the expectation and hence there is no approximation except the expansion itself. There’s a number of good examples of people doing this on Euler equations: Gali (2008, Chapter 2, Appendix 2.1) and Zietz (2006, Section 3.3) have both been useful for me personally.

Thanks for your reply. I understand that it’s perfectly fine to put it in dynare, because dynare has expectation operator as a default.

My question was, whether we can derive an analytical expression of P/D and state variables without the expectation operator, like what Bansal Yaron 2004 paper did. Do we need to assume a linear relationship like

\frac{P}{D} = A_0 + A_1 x

where x is the state variables? Or can we go directly from the recursive equation?Thanks

I’m not sure I understand - It sounds like you might be asking about the solution of a system of expectational difference equations (which is what the temporary equilibrium of a DSGE model is), which does express the model variables as functions of the states.

If that is indeed what you are asking about then there are many classic references: Blanchard and Kahn (1980), Sims (2001), and Schmitt-Grohe and Uribe (2004) would all be required reading. Or at least someone’s lecture notes on the topic.