# Log linearization

Consider some time series data $X=\{x_t:t\in[0,\infty)\}$. Define a steady state by \begin{align} x\in X:x_{t+1} - x_t = 0 \end{align}

and the log deviation from steady state with \begin{align} \hat x_t:=\ln x_t - \ln_x. \end{align}

Now we can express $x_t$ as \begin{align} x_t=x\exp(\hat x_t). \end{align}

Consider the the following equation \begin{align} a_t = b_t c_t^\alpha + D\mathbb{E}_t\left[\left(\frac{e_{t+1}}{e_t}\right)\left(\frac{f_t}{g_t}\right)^\beta a_{t+1}\right] \end{align} where $D$ is some constant, which is ought to be log linearized for every time dependent variable around steady state.

## First Attempt

Express every variable by its new definition,i.e. \begin{align} a\exp(\hat a_t) = b c^\alpha \exp(\hat b_t + \alpha\hat c_t)+ D\left(\frac{f}{g}\right)^\beta a\mathbb{E}_t\left[\exp(\hat e_{t+1} - \hat e_t + \beta(\hat f_t - \hat g_t) + \hat a_{t+1})\right] \end{align}

which should simplify to \begin{align} \exp(\hat a_t) = \exp(\hat b_t + \alpha\hat c_t)+\mathbb{E}_t\left[\exp(\hat e_{t+1} - \hat e_t + \beta(\hat f_t - \hat g_t) + \hat a_{t+1})\right] \end{align}

• Is this correct?
• What can be done to further improve/simplify the equation at hand?
• My answer to this question may be helpful. – cc7768 Jul 16 '15 at 23:30
• @cc7768 IMO your linked answer is close to being canonical. But this question here is probably the one that will mainly draw the attention of future askers on the matter (due it its more general title). Perhaps you would want to consider re-posting the answer here with any tweaks necessary, and then we could flag as duplicate any future question about log-linearization. – Alecos Papadopoulos Jul 17 '15 at 10:20
• @clueless Linearization involves a Taylor expansion. To perform a Taylor expansion, one has to specify with respect to which variables it should be calculated. Your equation has many time-varying quantities. Should the Taylor expansion be taken with respect to all of them? A subset of them? (In an intertemporal setting we usually consider Taylor expansions with respect to all the variables that have a steady-state value). Please include this information in the question. – Alecos Papadopoulos Jul 17 '15 at 10:25
• If I have time later today (got a co-author in town) then I'll try and reformulate that answer to be appropriate for this question. I agree this is the log-linearization question that will get more traffic. If someone else would like to answer this using some of my answer (or just their own answer) then that would make my life easier, but if no one has answered it in the next day or so then I will sit down and write something. – cc7768 Jul 17 '15 at 14:40
• Tried to clarify the exposition and gave a first solution hint. – clueless Jul 18 '15 at 13:33