This chapter is a collection of topics - not all related to each other - that you will probably find interesting or at least understandable, if you have read, and/ or feel comfortable with, the earlier chapter \ref{ch:solbase} on the basics of solving DSGE models. To provide at least some consistency, this chapter is divided into three sections. \textbf{The first section} deals directly with features of Dynare, such as dealing with correlated shocks, finding and saving your output, using loops, referring to external files and dealing with infinite eigenvalues. \textbf{The second section} overviews some of the inner workings of Dynare. The goal is to provide a brief explanation of the files that are created by Dynare to help you in troubleshooting or provide a starting point in case you actually want to customize the way Dynare works. \textbf{The third section} of the chapter focusses on modeling tips optimized for Dynare, but possibly also helpful for other work.\\
Other examples of .mod files used to generate impulse response functions are available on the Dynare website. In particular, Jesus Fernandez-Villaverde has provided a series of RBC model variants (from the most basic to some including variable capacity utilization, indivisible labor and investment specific technological change). You can find these, along with helpful notes and explanations, in the \href{http://www.cepremap.cnrs.fr/juillard/mambo/index.php?option=com_content&task=category§ionid=11&id=96&Itemid=89}{Official Examples} section of the Dynare website.\\
Also, don't forget to check occasionally the \href{http://www.cepremap.cnrs.fr/juillard/mambo/index.php?option=com_forum&Itemid=95}{Open Online Examples} page to see if any other user has posted an example that could help you in your work; or maybe you would like to post an example there yourself?
\subsection{Working with several correlated shocks}
Your model may have two or more shocks, and these may be correlated to each other. The example below illustrates how you would introduce this into Dynare. Actually, the example provided is somewhat more complete than strictly necessary. This is to give you an alternative, full-blown example to the one described in chapter \ref{ch:solbase}.
The economy consists of an infinitely living representative agent who values consumption $c_t$ and labor services $h_t$ according to the following utility function
where $i_t$ is investment and $y_t$ output. Consumers are therefore also owners of the firms. The economy is a real economy, where part of output can be consumed and part invested to form physical capital. As is standard, the law of motion of capital is given by
with $\alpha$ being the capital elasticity in the production function, with $0<\alpha<1$, and where $a_t$ represents a stochastic technological shock (or Solow residual). \\
Finally, we specify a \textbf{shock structure} that allows for shocks to display persistence across time and correlation in the current period. That is
where $|\rho+\tau|<1$ and $|\rho-\tau|<1$ to ensure stationarity (we call $\rho$ the coefficient of persistence and $\tau$ that of cross-persistence). Furthermore, we assume $\mathbb{E}_t (\epsilon_t)=0$, $\mathbb{E}_t (\nu_t)=0$ and that the contemporaneous variance-covariance matrix of the innovations $\epsilon_t$ and $\nu_t$ is given by
This system - probably quite similar to standard RBC models you have run into - yields the following first order conditions (which are straightforward to reproduce in case you have doubts\ldots) and equilibrium conditions drawn from the description above. Note that the first equation captures the labor supply function and the second the intertemporal consumption Euler equation.
To ``translate'' the model into a language understandable by Dynare, we would follow the steps outlined in chapter \ref{ch:solbase}. We will assume that you're comfortable with these and simply present the final .mod file below. Fist, though, note that to introduce shocks into Dynare, we have two options (this was not discussed in the earlier chapter). Either write:\\
\\
\texttt{shocks;\\
var e; stderr 0.009;\\
var u; stderr 0.009;\\
var e, u = phi*0.009*0.009;\\
end;}\\
\\
where the last line specifies the contemporaneous correlation between our two exogenous variables. (** does this mean every time you run your .mod file you will get a slightly different shock?) \\
Alternatively, you can also write: \\
\\
\texttt{shocks;\\
var e = 0.009 $\hat{}$ 2;\\
var u = 0.009 $\hat{}$ 2;\\
var e, u = phi*0.009*0.009;\\
end;}\\
So that you can gain experience by manipulating the entire model, here is the complete .mod file corresponding to the above example:\\
To save your simulated variables, you can add the following command at the end of your .mod file: \texttt{dynasave (FILENAME) [variable names separated by commas]} If no variable names are specified in the optional field, Dynare will save all endogenous variables. In Matlab, variables saved with the \texttt{dynasave} command can be retrieved by using the Matlab command \texttt{load -mat FILENAME}.
You may find it convenient to refer to an external file, either to compute the steady state of your model, or when specifying shocks in an external file. The former is described in section \ref{sec:ssshock} of chapter \ref{ch:solbase} when discussing steady states. The advantage of using Matlab, say, to find your model's steady state was clear with respect to Dynare version 3, as the latter resorted to numerical approximations to find steady state values. But Dynare version 4 now uses the same analytical methods available in Matlab. For most usage scenarios, you should therefore do just as well to ask Dynare to compute your model's steady state (except, maybe, if you want to run loops, to vary your parameter values, for instance, in which case writing a Matlab program may be more handy). (** give more details? an example?)\\
But you may also be interested in the second possibility described above, namely of specifying shocks in an external file, to simulate a model based on shocks from a prior estimation, for instance. You could then retrieve the exogenous shocks from the oo\_ file by saving them in a file called datafile.mat. Finally, you could simulate a deterministic model with the shocks saved from the estimation by specifying the source file for the shocks, using the \\\mbox{\texttt{shocks(shocks\_file = datafile.mat)}} command. (** how should the variables be ordered in this datafile?)
But of course, this is a bit of a workaround, since you could also use the built-in commands in Dynare to generate impulse response functions from estimated shocks, as described in chapter \ref{ch:estbase}. \\
If you use the command \texttt{check} in your .mod file, Dynare will report your system's eigenvalues and tell you if these meet the Blanchard-Kahn conditions. At that point, don't worry if you get infinite eigenvalues - these are are firmly grounded in the theory of generalized eigenvalues. They have no detrimental influence on the solution algorithm. As far as Blanchard-Kahn conditions are concerned infinite eigenvalues are counted as explosive roots of modulus larger than one. \\
\section{Files created by Dynare}
At times, you may get a message that there is an error in a file with a new name, or you may want to have a closer look at how Dynare actually solves your model - out of curiosity or maybe to do some customization of your own. To help address these issues, below is a brief description of which files are created by Dynare after you launch \texttt{simul} or \texttt{stoch\_simul}, what their purpose is and what they contain. \\
Models in Dynare must be stationary, such that you can linearize them around a steady state and return to steady state after a shock. Thus, you must first stationarize your model, then linearize it, either by hand, or by letting Dynare do the work. You can then reconstruct ex-post the non-stationary simulated variables after running impulse response functions.\\
For deterministic models, the trick is to use only stationary variables in $t+1$. More generally, if $y_t$ is $I(1)$, you can always write $y_{t+1}$ as $y_t+dy_{t+1}$, where $dy_t= y_t-y_{t-1}$. Of course, you need to know the value of $dy_t$ at the final equilibrium.\\
Note that in a stationary model, it is expected that variables will eventually go back to steady state after the initial shock. If you expect to see a growing curve for a variable, you are thinking about a growth model. Because growth models are nonstationary, it is easier to work with the stationarized version of such models. Again, if you know the trend, you can always add it back after the simulation of the stationary components of the variables.
which can be written as a recursive system of the form:
\[S1_t=x_t+\beta S1_{t+1},\]
\[S2_t=y_t+\gamma S2_{t+1},\]
\[S1=p_t S2.\]
This is particularly helpful, for instance, in a \textbf{Calvo type setting}, as illustrated in the following brief example. The RBC model with monopolistic competition introduced in chapter \ref{ch:solbase} involved flexible prices. The extension with sticky prices, <20> la Calvo for instance, is instead typical of the new Keynesian monetary literature, exemplified by papers such as \citet{ClaridaGaliGertler1999}. \\
The optimal price for a firm resetting its price in period $t$, given that it will be able to reset its price only with probability $1-\theta$ each period, is
where $\mu$ is the markup, $\beta$ is a discount factor, $i$ represents a firm of the continuum between 0 and 1, and $mc_t$ is marginal cost as described in the example in chapter \ref{ch:solbase}. The trouble, of course, is \textbf{how to input this infinite sum into Dynare}? \\
It turns out that the Calvo price setting implies that the aggregate price follows the equation of motion $p_t =\theta p_{t-1}+(1-\theta) p_t^*$, thus implying the following inflation relationship $\pi_t =(1-\theta)(p_t^*- p_{t-1})$. Finally, we can also rewrite the optimal price setting equation, after some algebraic manipulations, as
where $\widehat{mc}_{t+k}= mc_{t+k}+\mu$ is the deviation of the marginal cost from its natural rate, defined as the marginal cost when prices are perfectly flexible. \\
The trick now is to note that the above can be written recursively, by writing the right hand side as the first term of the sum (with $k=0$) plus the remainder of the sum, which can be written as the left hand side term scrolled forward one period and appropriately discounted. Mathematically, this yields:
where $\lambda\equiv\frac{(1-\theta)(1-\beta\theta)}{\theta}$, which is the recognizable inflation equation in the new Keynesian (or new Neoclassical) monetary literature.
\subsection{Infinite sums with changing timing of expectations}
When you are not able to write an infinite sum recursively, as the index of the expectations changes with each element of the sum, as in the following example, a different approach than the one mentioned above is necessary. \\
where $y_t$ and $x_t$ are endogenous variables. \\
In Dynare, the best way to handle this is to write out the first $k$ terms explicitly and enter each one in Dynare, such as: $\mathbb{E}_{t-1}x_t +\mathbb{E}_{t-2}x_t+\ldots+\mathbb{E}_{t-k}x_t$.
