Log-Linearization of a White Noise Process


I would be grateful if someone could help me with the following; in my model, I am imposing a Taylor rule that is of the following form:

where “R” is the net policy rate, “pi” is gross inflation, “y” is gross rate of output growth, and these small "r"s are coefficients in the Taylor rule.

My question is: since I have log-linearized by hand the rest of the model, but would like to have the monetary policy shock, is there a way to log-linearize the above form of the Taylor rule given that “epsilon_R” is a white noise process (hence it’s SS value is 0)?

For instance, would it be possible to express all the other variables as log-deviations from their SS values whilst leaving the monetary policy shock enter additively just like in the above equation and then set its variance in the “parameters” section so as to match some empirical counterpart?

Or, would it be acceptable to assume there is no monetary policy shock to begin with, then log-linearize the above equation, and then at the end suppose there exists a shock that enters additively in the log-linearized version of the Taylor rule?

Another possibility (I think) would be to have a multiplicative shock with mean 1 that multiplies the whole of the RHS in the Taylor rule where there is no additive shock, but I would rather avoid this (if possible).

Thank you very much.

To add to my previous post: the log-linear form of the Taylor rule above (without the monetary policy shock) should be given by (1) below. Since “R” is positive in the SS, I take R^hat to mean log(R_t) - log® rather than log(1+R_t) - log(1+R).

I also wrote down (2) which should, when log-linearized, give exactly (1) plus an additive shock expressed as a log-deviation from its SS. Would this be a possible solution?

Thank you.

I am not sure I am following. Your problem seems to be that you want to impose a linear Taylor rule that is in levels of some variables instead of log deviations of levels from their steady state. Correct? In that case, doing the linearization as in (1) is correct (if you assume that pi=y=1 in steady state, otherwise, you are missing those). You simply need to add the shock. The way to derive this equation is to do a first order Taylor approximation in all the variables and shocks and then expand the fractions to get percentage deviations. E.g.

R_t \approx 1*(R_t-R)=R*(R_t-R)/R=R* \hat R_t

Only for the last term, where the steady state is 0, you do not expand the fraction.

Prof. Pfeifer,

Thank you very much for your reply. I’m sorry for not being a bit more clear - I will restate briefly my problem below.

  1. I have log-linearized all of my model equations by hand and solved for the steady state.

  2. All the interest rates in my model are on a net basis. Since none are zero in the SS, I have taken R^hat to mean: R^hat = log(R_t) - log®. Please let me know if this is not advisable. I tried doing it otherwise, namely defining R^hat = log(1+R_t) - log(1+R) but then, I had to rely on the approximation (I am not sure if this approximation is valid and innocuous)

(1+R_t)^hat \approx (R/(1+R))*R^hat

which I would rather avoid as it often reported some errors as I varied the parameters (perhaps due to an algebraic mistake on my part), but this was not the case when I used the Taylor rule similar to (2).

  1. So, I would like to introduce a Taylor rule that is stated in terms of the net interest rates to avoid relying on the above approximation. So one particular example was the originally stated Taylor rule, but without the shock

where “pi_t” is gross inflation and is 1 in SS by assumption, whilst “y_t”, gross output growth, should be 1 in SS by definition. By substituting in for all variables their equivalent representation involving their SS value and their log-deviation from their SS (say, plugging y_t = y*(1+hat(y_t))), this Taylor rule (without the shock) should I think be transformable into equation (1).

4. So my question is: is my suggestion numbered (2) in my previous post a valid way to get what I want, namely get equation (1) plus an exogenous shock? I am looking for a Taylor rule that involves the net policy rate, contains a shock, and can be log-linearized. That’s all.

  1. If the answer to 4. is yes, then one final question: should the shock “epsilon_{R,t}” be of mean 1 or mean 0? If it should be mean zero, then one is unable (I think) to take

“epsilon_{R,t}”^hat = log(epsilon_{R,t}) - log(epsilon_{R})

since the second term is not defined.

For instance, in Iacoviello (2005), the following Taylor rule is used:

So how is Iacoviello treating the monetary policy shock in his log-linearization of the Taylor rule?

Apologies if what I am asking is trivial, but I have very little experience regarding these issues.

Your previous equation (2) needs to have an

instead of just

Otherwise, in steady state the interest rate would be 0. Also, I don’t see where the exponent R comes from. You should have

Regarding Iacoviello (2005): that statement on epsilon is wrong. Either there is an exp() missing in his equation (13) or the epsilon is mean 1.