Log linearize forward looking variable manually?


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.