This chapter covers everything that leads to, and stems from, the solution of DSGE models; a vast terrain. That is to say that the term ``solution'' in the title of the chapter is used rather broadly. You may be interested in simply finding the solution functions to a set of first order conditions stemming from your model, but you may also want to go a bit further. Typically, you may be interested in how this system behaves in response to shocks, whether temporary or permanent. Likewise, you may want to explore how the system comes back to its steady state or moves to a new one. This chapter covers all these topics. But instead of skipping to the topic closest to your needs, we recommend that you read this chapter chronologically, to learn basic Dynare commands and the process of writing a proper .mod file - this will serve as a base to carry out any of the above computations.
\section{An important distinction}\label{sec:distcn}
Before speaking of Dynare, it is important recognize a distinction in model types. This distinction will appear throughout the chapter; in fact, it is so fundamental, that we considered writing separate chapters altogether. But the amount of common material - Dynare commands and syntax - is notable and writing two chapters would have been overly repetitive. Enough suspense; here is the important distinction: \textbf{is your model stochastic or deterministic?}\\
The distinction hinges on \textbf{whether future shocks are known}. In deterministic models, the occurrence of all future shocks is known exactly at the time of computing the model's solution. In stochastic models, instead, only the distribution of future shocks is known. Thus, deterministic models can be solved exactly. Stochastic models must be approximated. Intuitively, if an agent has perfect foresight, she can specify today what each of her precise actions will be in the future. But when the future has a random component to it, the best the agent can do is specify a decision rule for the future: what will her optimal action be contingent on each possible realization of a shock. It is therefore much harder to solve a stochastic model than a deterministic one: in the latter case we look for a series of numbers that match a set of given equations, whereas in the former, we search for a function which, to complicate things, may be non-linear and thus needs to be approximated. In control theory, the latter is usually called a ``closed loop'' solution, and the former an ``open loop''.\\
Because this distinction will resurface again and again throughout the chapter, but also because it has been a source of significant confusion in the past, the following gives some additional details.
\subsection{\textsf{NOTE!} Deterministic vs stochastic models}\label{sec:detstoch}
\textbf{Deterministic} models have the following characteristics:
\begin{enumerate}
\item As the DSGE (read, ``stochastic'', i.e. not deterministic!) literature has gained attention in economics, deterministic models have become somewhat rare. Examples include OLG models without aggregate uncertainty.
\item These models are usually introduced to study the impact of a change in regime, as in the introduction of a new tax, for instance.
\item Models assume full information, perfect foresight and no uncertainty around shocks.
\item Shocks can hit the economy today or at any time in the future, in which case they would be expected with perfect foresight. They can also last one or several periods.
\item Most often, though, models introduce a positive shock today and zero shocks thereafter (with certainty).
\item The solution does not require linearization, in fact, it doesn't even really need a steady state. Instead, it involves numerical simulation to find the exact paths of endogenous variables that meet the model's first order conditions and shock structure.
\item This solution method can therefore be useful when the economy is far away from steady state (when linearization offers a poor approximation).
\end{enumerate}
\textbf{Stochastic} models, instead, have the following characteristics:
\begin{enumerate}
\item These types of models tend to be more popular in the literature. Examples include most RBC models, or new keynesian monetary models.
\item In these models, shocks hit today (with a surprise), but thereafter their expected value is zero. Expected future shocks, or permanent changes in the exogenous variables cannot be handled due to the use of Taylor approximations around a steady state.
\item Note that when these models are linearized to the first order, agents behave as if future shocks where equal to zero (since their expectation is null), which is the \textbf{certainty equivalence property}. This is an often overlooked point in the literature which misleads readers in supposing their models may be deterministic.
The goal of this first section is to introduce a simple example. Future sections will aim to code this example into Dynare and analyze its salient features under the influence of shocks - both in a stochastic and a deterministic environment. Note that as a general rule, the examples in the basic chapters, \ref{ch:solbase} and \ref{ch:estbase}, are kept as bare as possible, with just enough features to help illustrate Dynare commands and functionalities. More complex examples are instead presented in the advanced chapters.\\
The model introduced here is a basic RBC model with monopolistic competition, used widely in the literature and taken, in its aspect presented below, from notes available on Jesus Fernandez-Villaverde's very instructive \href{http://www.econ.upenn.edu/~jesusfv/}{website}; this is a good place to look for additional information on any of the following model set-up and discussion.\\
The above equation can be seen as an accounting identity, with total expenditures on the left hand side and revenues - including the liquidation value of the capital stock - on the right hand side. Alternatively, with a little more imagination, the equation can also be interpreted as a capital accumulation equation after bringing $c_t$ to the right hand side and noticing that $w_t l_t + r_t k_t$, total payments to factors, equals $y_t$, or aggregate output, by the zero profit condition. As a consequence, if we define investment as $i_t=y_t - c_t$, we obtain the intuitive result that $i_t=k_{t+1}-(1-\delta) k_{t}$, or that investment replenishes the capital stock thereby countering the effects of depreciation. In any given period, the consumer therefore faces a tradeoff between consuming and investing in order to increase the capital stock and consuming more in following periods (as we will see later, production depends on capital).\\
Maximization of the household problem with respect to consumption, leisure and capital stock, yields the Euler equation in consumption, capturing the intertemporal tradeoff mentioned above, and the labor supply equation linking labor positively to wages and negatively to consumption (the wealthier, the more leisure due to the decreasing marginal utility of consumption). These equation are
The firm side of the problem is slightly more involved, due to monopolistic competition, but is presented below in the simplest possible terms, with a little hand-waiving involved, as the derivations are relatively standard. \\
There are two ways to introduce monopolistic competition. We can either assume that firms sell differentiated varieties of a good to consumers who aggregate these according to a CES index. Or we can postulate that there is a continuum of intermediate producers with market power who each sell a different variety to a competitive final goods producer whose production function is a CES aggregate of intermediate varieties.\\
If we follow the second route, the final goods producer chooses his or her optimal demand for each variety, yielding the Dixit-Stiglitz downward sloping demand curve. Intermediate producers, instead, face a two pronged decision: how much labor and capital to employ given these factors' perfectly competitive prices and how to price the variety they produce.\\
Production of intermediate goods follows a CRS production function defined as
where the $i$ subscript stands for firm $i$ of a continuum of firms between zero and one and where $\alpha$ is the capital elasticity in the production function, with $0<\alpha<1$. Also, $z_t$ captures technology which evolves according to
\[
z_t = \rho z_{t-1} + e_t
\]
where $\rho$ is a parameter capturing the persistence of technological progress and $e_t \thicksim\mathcal{N}(0,\sigma)$. \\
where $p_{it}$ is firm $i$'s specific price, $mc_t$ is real marginal cost and $p_t$ is the aggregate CES price or average price. An additional step simplifies this expression: symmetric firms implies that all firms charge the same price and thus $p_{it}=p_t$; we therefore have: $mc_t =(\epsilon-1)/\epsilon$\\
But what are marginal costs equal to? To find the answer, we combine the optimal capital to labor ratio into the production function and take advantage of its CRS property to solve for the amount of labor or capital required to produce one unit of output. The real cost of using this amount of any one factor is given by $w_tl_{it}+ r_tk_{it}$ where we substitute out the payments to the other factor using again the optimal capital to labor ratio. When solving for labor, for instance, we obtain
which does not depend on $i$; it is thus the same for all firms. \\
Interestingly, the above can be worked out, by using the optimal capital to labor ratio, to yield $w_t [(1-\alpha)y_{it}/l_{it}]^{-1}$, or $w_t \frac{\partial l_{it}}{\partial y_{it}}$, which is the definition of marginal cost: the cost in terms of labor input of producing an additional unit of output. This should not be a surprise since the optimal capital to labor ratio follows from the maximization of the production function (minus real costs) with respect to its factors. \\
Combining this result for marginal cost, as well as its counterpart in terms of capital, with the optimal pricing condition yields the final two important equations of our model
To end, we aggregate the production of each individual firm to find an aggregate production function. On the supply side, we factor out the capital to labor ratio, $k_t/l_t$, which is the same for all firms and thus does not depend on $i$. On the other side, we have the Dixit-Stiglitz demand for each variety. By equating the two and integrating both side, and noting that price dispersion is null - or that, as hinted earlier, $p_{it}=p_t$ - we obtain aggregate production
\[
y_t = A_t k_t^\alpha l_t^{1-\alpha}
\]
which can be shown is equal to the aggregate amount of varieties bought by the final good producer (according to a CES aggregation index) and, in turn, equal to the aggregate output of final good, itself equal to household consumption. Note, to close, that because the ratio of output to each factor is the same for each intermediate firm and that firm specific as well as aggregate production is CRS, we can rewrite the above two equations for $w_t$ and $r_t$ without the $i$ subscripts on the right hand side. \\
This ends the exposition of the example. Now, let's roll up our sleeves and see how we can input the model into Dynare and actually test how the model will respond to shocks.
Input into Dynare involves the .mod file, as mentioned loosely in the introduction of this Guide. The .mod file can be written in any editor, external or internal to Matlab. It will then be read by Matlab by first navigating within Matlab to the directory where the .mod file is stored and then by typing in the Matlab command line \texttt{Dynare filename.mod;} (although actually typing the extension .mod is not necessary). But before we get into executing a .mod file, let's start by writing one! \\
The preamble generally involves three commands that tell Dynare what are the model's variables, which are endogenous and what are the parameters. The \textbf{commands} are:
\begin{itemize}
\item\texttt{var} starts the list of endogenous variables, to be separated by commas.
\item\texttt{varexo} starts the list of exogenous variables that will be shocked.
\item\texttt{parameters} starts the list of parameters and assigns values to each.
\end{itemize}
In the case of our example, here's what the \textbf{preamble would look like}:\\
As you can tell, writing a .mod file is really quite straightforward. Two quick comments: \textsf{\textbf{NOTE!}} Remember that each instruction of the .mod file must be terminated by a semicolon (;), although a single instruction can span two lines if you need extra space (just don't put a semicolon at the end of the first line). \textsf{\textbf{TIP!}} You can also comment out any line by starting the line with two forward slashes (//), or comment out an entire section by starting the section with /* and ending with */. For example:\\
\\
\texttt{var y c k i l y\_l w r z;\\
varexo e;\\
parameters beta psi delta \\
alpha rho sigma epsilon;\\
// the above instruction reads over two lines\\
/*\\
the following section lists\\
several parameters which were\\
calibrated by my co-author. Ask\\
her all the difficult questions!\\
*/\\
alpha = 0.33;\\
beta = 0.99;\\
delta = 0.023;\\
psi = 1.75;\\
rho = 0.95; \\
sigma = (0.007\/(1-alpha));\\
epsilon = 10;}\\
\\
A final \textsf{\textbf{TIP!}}: Dynare accepts standard Matlab expressions in any of the model equations. The only thing you need to do is to start the line with \# and then enter the Matlab command as you would normally. This is useful if you want to define a parameter in terms of another; for instance, you can write \texttt{\# b = 1 / c}. This feature, actually, is more useful when carrying out model estimation.
One of the beauties of Dynare is that you can \textbf{input your model's equations naturally}, almost as if you were writing them in an academic paper. This greatly facilitates the sharing of your Dynare files, as your colleagues will be able to understand your code in no-time. There are just a few conventions to follow. Let's first have a look at our \textbf{model in Dynare notation}, and then go through the various Dynare input conventions. What you can already try to do is glance at the model block below and see if you can recognize the equations in the earlier example. See how easy it is to read Dynare code? \\
Just in case you need a hint or two to recognize these equations, here's a brief description: the first equation is the Euler equation in consumption. The second the labor supply function. The third the accounting identity. The fourth is the production function. The fifth and sixth are the marginal cost equal to markup equations. The seventh is the investment equality. The eighth an identity that may be useful and the last the equation of motion of technology.
The above example illustrates the use of a few important commands and conventions to translate a model into a Dynare-readable .mod file.
\begin{itemize}
\item The first thing to notice, is that the model block of the .mod file begins with the command \texttt{model} and ends with the command \texttt{end}.
\item Second, in between, there need to be as many equations as you declared endogenous variables (this is actually one of the first things that Dynare checks; it will immediately let you know if there are any problems).
\item Third, as in the preamble and everywhere along the .mod file, each line of instruction ends with a semicolon (except when a line is too long and you want to break it across two lines. This is unlike Matlab where if you break a line you need to add \ldots).
\item Fourth, equations are entered one after the other; no matrix representation is necessary. Note that variable and parameter names used in the model block must be the same as those declared in the preamble; \textsf{\textbf{TIP!}} remember that variable and parameter names are case sensitive.
\item Variables entering the system with a time $t$ subscript are written plainly. For example, $x_t$ would be written $x$.
\item Variables entering the system with a time $t-n$ subscript are written with $(-n)$ following them. For example, $x_{t-2}$ would be written $x(-2)$ (incidentally, this would count as two backward looking variables).
\item In the same way, variables entering the system with a time $t+n$ subscript are written with $(+n)$ following them. For example, $x_{t+2}$ would be written $x(+2)$. Writing $x(2)$ is also allowed, but this notation makes it harder to count the number of forward looking variables; more on this below \ldots
\item In Dynare, the timing of each variable reflects when that variable is decided. For instance, our capital stock is not decided today, but yesterday (recall that it is a function of yesterday's investment and capital stock); it is what we call in the jargon a \textbf{predetermined} variable. Thus, eventhough in the example presented above we wrote $k_{t+1}=i_t +(1-\delta)k_t$, we would write it in Dynare as \texttt{k=i+(1-delta)*k(-1)}.
\item As another example, consider that in some wage negociation models, wages used during a period are set the period before. Thus, in the equation for wages, you can write wage in period $t$ (when they are set), but in the labor demand equation, wages should appear with a one period lag.
\item A slightly more roundabout way to explain the same thing is that for stock variables, you must use a ``stock at the end of the period'' concept. It is investment during period $t$ that sets stock at the end of period $t$. Be careful, a lot of papers use the ``stock at the beginning of the period'' convention, as we did (on purpose to highlight this distinction!) in the setup of the example model above.
\item A (+1) next to a variable tells Dynare to count the occurrence of that variable as a jumper or forward-looking or non-predetermined variable.
\item\textbf{Blanchard-Kahn} conditions are met only if the number of non-predetermined variables equals the number of eigenvalues greater than one. If this condition is not met, Dynare will put up a warning.
\item Note that a variable may occur both as predetermined and non-predetermined. For instance, consumption could appear with a lead in the Euler equation, but also with a lag in a habit formation equation. In this case, the second order difference equation would have two eigenvalues, one needing to be greater and the other smaller than one for stability.
There are two other variants of the system's equations which Dynare accommodates. First, the \textbf{linear model} and second, the \textbf{model in exp-logs}. In the first case, all that is necessary is to write the term \texttt{(linear)} next to the command \texttt{model}. Our example, with just the equation for $y_l$ for illustration, would look like:\\
\\
\texttt{model (linear);\\
y*\_l=y* - l*;\\
end;}\\
\\
where a star next to a variable means difference from steady state.\\
Otherwise, you may be interested to have Dynare take Taylor series expansions in logs rather than in levels; this turns out to be a very useful option when estimating models with unit roots, as we will see in chapter \ref{ch:estbase}. If so, simply rewrite your equations by taking the exponential and logarithm of each variable. The Dynare input convention makes this very easy to do. Our example would need to be re-written as follows (just shown for the first two equations)\\
\section{Specifying transitions (steady states and shocks)}\label{sec:ssshock}
A ``transition'' usually entails two things: starting from somewhere and ending up somewhere else! In the more mathematical language of modern economics, this usually corresponds to an original steady state and a shock, and at times also a final (different) steady state. The combination of steady states and shocks will vary according to whether you're dealing with a deterministic or a stochastic model (remember the difference? if not, have a quick look at section \ref{sec:distcn} above). We will make this clear by introducing appropriate subsection headings below. \\
\subsection{Specifying steady states - introduction}
To specify steady states, the appropriate commands are \texttt{initval}, \texttt{endval} or, more rarely, \texttt{histval} which is covered only in the \href{http://www.cepremap.cnrs.fr/juillard/mambo/index.php?option=com_content&task=view&id=51&Itemid=84}{Reference Manual}. Here, we first dwell on the \texttt{initval} command and later come back to the \texttt{endval} command which is used only in certain circumstances. \\
The \texttt{initval} command is used to declare your model's starting values, which actually don't necessarily need to be steady state values. This opens the door to several possibilities. You can provide exact or approximate steady state values, or you can simply specify an arbitrary point from which to start your simulations. Let's be a little more concrete and analyze each of these cases.\\
\textbf{If the command \texttt{initival} is followed by \texttt{steady}}, Dynare will understand that simulations or impulse response functions should begin from the steady state. In addition, Dynare will check to see if the values you provided exactly match or only approximate the steady state. In the latter case, Dynare will replace your guesstimates with the model's true steady state values. In our example, this would look like:\\
Note that the \texttt{initval} block is closed with \texttt{end}. Within the block you should list values for all your variables - endogenous and exogenous (even if the latter is not required for stochastic models, but you may want to get in the habit of doing so anyhow).\\
\textbf{If the command \texttt{initival} is not followed by \texttt{steady}}, Dynare will start the simulations or impulse response function from whatever values you provide. This means that if you provided exact steady state values, Dynare will analyze how shocks will move your system away from steady state. But if you provided other values, Dynare will calculate the steady state anyhow, to be used when linearizing the model as in the case of a stochastic model, but will begin simulations or impulse response functions from the values you entered. \textsf{\textbf{TIP!}} If you're dealing with a stochastic model, remember that its linear approximation is good only in the vicinity of the steady state, thus we strongly recommend that you start your simulations from a steady state; this means either using the command \texttt{steady} or entering exact steady state values. \\
The difficulty in the above, of course, is calculating the actual steady state values. Doing so borders on a form of art, and luck is unfortunately part of the equation. Yet, the following \textsf{\textbf{TIPS!}} may help.\\
As mentioned above, Dynare can help in finding your model's steady state by calling the appropriate Matlab functions. But it is usually only successful if the initial values you entered are close to the true steady state. If you have trouble finding the steady state of your model, you can begin by playing with the \textbf{options following the \texttt{steady} command}. These are:
For complicated models, finding good numerical initial values for the endogenous variables is the trickiest part of finding the equilibrium of that model. Often, it is better to start with a smaller model and add new variables one by one.\\
But even for simpler models, you may still run into difficulties in finding your steady state. If so, another option is to \textbf{enter your model in linear terms}. In this case, variables would be expressed in percent deviations from steady state. Thus, their initial values would all be zero. Unfortunately, if any of your original (non-linear) equations involve sums (a likely fact), your linearized equations will include ratios of steady state values, which you would still need to calculate. Yet, you may be left needing to calculate fewer steady state values than in the original, non-linear, model. \\
Alternatively, you could also use an \textbf{external program to calculate exact steady state values}. For instance, you could write an external Maple file and then enter the steady state solution by hand in Dynare. But of course, this procedure could be time consuming and bothersome, especially if you want to alter parameter values (and thus steady states) to undertake robustness checks. In that case, you could write a Matlab program to find your model's steady state. Doing so has two clear advantages: first, Matlab is able to find steady states analytically - a procedure that can be more effective than numerical estimation. And second, you can incorporate a Matlab program directly into your .mod file so that running loops with different parameter values, for instance, becomes seamless. When doing so, your matlab (.m) file should have the same name as your .mod file, followed by \texttt{\_steadystate} For instance, if your .mod file is called \texttt{example.mod}, your Matlab file should be called \texttt{example\_steadystate.m} Note that Dynare will automatically check the directory where you've saved your .mod file to see if such a Matlab file exists. If so, it will use that file to find steady state values regardless of whether you've provided initial values in your .mod file or used the command \texttt{steady}. (** provide example of .m file below)\\
\caption[Shock and model-type matrix]{Depending on the model type you're working with and the desired shocks, you will need to mix and match the various steady state and shock commands.}
When working with a deterministic model, you have the choice of introducing both temporary and permanent shocks. One thing to note, though, is that a deterministic model is written somewhat differently than our earlier example. The only difference is that we no longer need the $e_t$ variable, as we can make $z_t$ directly exogenous. That is, we can replace $e_t$ with $z_t$ in the \texttt{varexo} instruction line, and delete $z_t$ from the \texttt{var} line. In the model block we can delete the last equation corresponding to the motion of technology. This is because in a deterministic model, the shock affects the exogenous variable directly. Let's see how \ldots<EFBFBD>\\
To work with a \textbf{temporary shock}, you are free to set the duration and level of the shock. To specify a shock that lasts 9 periods on $e_t$, for instance, you would write:\\
Given the above instructions, Dynare would replace the value of $z_t$ specified in the \texttt{initval} block with the value of 0.1 entered above. If variables were in logs, this would have corresponded to a 10\% shock. Note that you can also use the \texttt{mshock} command which multiplies the initial value of an exogenous variable by the \texttt{mshock} value (** is this feature still in version 4?). Finally, note that we could have entered future periods in the shocks block, such as \texttt{periods 5:10}, in order to simulate an \textbf{expected temporary} shock. You could also increase the terminal period to the maximum scale of the simulation, in order to introduce an \textbf{expected permanent} shock. (** is this true?)\\
To study the effects of a \textbf{permanent shock} hitting the economy today, such as a structural change in your model, you would not specify actual ``shocks'', but would simply tell the system to which steady state values you would like it to move to and let Dynare calculate the transition path. To do so, you would use the \texttt{endval} block following the usual \texttt{initval} block. For instance, you may specify all values to remain common between the two blocks, except for the value of technology which presumably changes permanently. The corresponding instructions would be:\\
where \texttt{steady} can also be added to the \texttt{endval} block. Doing so has the same effect as mentioned earlier, namely of computing the steady state value for endogenous variables using the information given in the \texttt{endval} block. Specifically, the values given for endogenous variables are used as initial guess for computation, and the values given for exogenous variables are kept constant over time. In our example case, we make use of the second \texttt{steady} since the terminal steady state values are bound to be somewhat different from those in the original steady state after changing the value of technology.\\
Recall from our earlier description of stochastic models that shocks are only allowed to be temporary. A permanent shock cannot be accommodated due to the need to stationarize the model around a steady state. Furthermore, shocks can only hit the system today, as the expectation of future shocks must be zero. With that in mind, we can however make the effect of the shock propagate slowly throughout the economy by introducing a ``latent shock variable'' such as $e_t$ in our example, that affects the model's true exogenous variable, $z_t$ in our example, which is itself an $AR(1)$, exactly as in the model we introduced from the outset. In that case, though, we would declare $z_t$ as an endogenous variable and $e_t$ as an exogenous variable, as we did in the preamble of the .mod file in section \ref{sec:preamble}. Supposing we wanted to add a shock with variance $\sigma^2$, where $\sigma$ is determined in the preamble block, we would write: \\
Two last things to note: first, you can actually \textbf{mix in deterministic shocks} in stochastic models by using the commands \texttt{varexo\_det} and listing some shocks as lasting more than one period in the \texttt{shocks} block. For information on how to do so, please see the \href{http://www.cepremap.cnrs.fr/juillard/mambo/index.php?option=com_content&task=view&id=51&Itemid=84}{Reference Manual}.\\
Second, a handy command that you can add after the \texttt{initval} or \texttt{endval} block (following the \texttt{steady} command if you decide to add one) is the \texttt{check} command. This \textbf{computes and displays the eigenvalues of your system} which are used in the solution method. As mentioned earlier, a necessary condition for the uniqueness of a stable equilibrium in the neighborhood of the steady state is that there are
as many eigenvalues larger than one in modulus as there are forward looking variables in the system. If this condition is not met, Dynare will tell you that the Blanchard-Kahn conditions are not satisfied (whether or not you insert the \texttt{check} command). \\
\section{Selecting a computation}\label{sec:compute}
So far, we have written an instructive .mod file, but what should Dynare do with it? What are we interested in? In most cases, it will be impulse response functions (IRFs) due to the external shocks. Let's see which are the appropriate commands to give to Dynare. Again, we will distinguish between deterministic and stochastic models. \\
In the deterministic case, all you need to do is add the command \texttt{simul} at the bottom of your .mod file. Note that the command takes the option \mbox{\texttt{[ (periods=INTEGER) ] }} The command \texttt{Simul} triggers the computation of a deterministic simulation of the model for the number of periods set in the option. To do so, it uses a Newton method to solve simultaneously all the equations for every period (see \citet{Juillard1996} for details). The simulated variables are available in global matrix y\_ in which variables are arranged row by row, in alphabetical order. \\
In the more common case of stochastic models, the command \texttt{stoch\_simul} is appropriate. This command instructs Dynare to compute a Taylor approximation of the decision and transition functions for the model (the equations listing current values of the endogenous variables of the model as a function of the previous state of the model and current shocks), impulse response
functions and various descriptive statistics (moments, variance decomposition, correlation and autocorrelation coefficients).\footnote{For correlated shocks, the variance decomposition is computed as in the VAR literature through a Cholesky
decomposition depends upon the order of the variables in the varexo command. (** make sure it depends on the order of variable declaration and not the alphabetical order. Also, is it possible to declare the correlation between shocks in a triangular representation?).} If you're interested to peer a little further into what exactly is going on behind the scenes of Dynare's computations, have a look at Chapter \ref{ch:solbeh}. Here instead, we focus on the application of the command and reproduce below the table of options that can be added to \texttt{stoch\_simul}. (** maybe just include a subset and refer to Ref Man for complete set? ie is the following useful as is, or too long?) \\
\item ar = INTEGER: Order of autocorrelation coefficients to compute and to print (default = 5).
\item dr\_algo = 0 or 1: specifies the algorithm used for computing the quadratic approximation of the decision rules: $0$ uses a pure perturbation approach as in \citet{SchmittGrohe2004} (default) and 1 moves the point around which the Taylor expansion is computed toward the means of the distribution as in \citet{CollardJuillard2001a}.
\item drop = INTEGER: number of points dropped at the beginning of simulation before computing the summary
statistics (default = 100).
\item hp\_filter = INTEGER: uses HP filter with lambda = INTEGER before computing moments (default: no filter).
\item hp\_ngrid = INTEGER: number of points in the grid for the discreet Inverse Fast Fourier Transform used in the
HP filter computation. It may be necessary to increase it for highly autocorrelated processes (default = 512).
\item irf = INTEGER: number of periods on which to compute the IRFs (default = 40). Setting IRF=0, suppresses the
plotting of IRF<52>s.
\item relative\_irf requests the computation of normalized IRFs in percentage of the standard error of each shock.
\item nocorr: doesn<73>t print the correlation matrix (printing is the default).
\item nofunctions: doesn<73>t print the coefficients of the approximated solution (printing is the default).
\item nomoments: doesn<73>t print moments of the endogenous variables (printing them is the default).
\item noprint: cancel any printing; usefull for loops.
\item order = 1 or 2 : order of Taylor approximation (default = 2), unless you're working with a linear model in which case the order is automatically set to 1.
\item periods = INTEGER: specifies the number of periods to use in simulations (default = 0).
\item qz\_criterium = INTEGER or DOUBLE: value used to split stable from unstable eigenvalues in reordering the
Generalized Schur decomposition used for solving 1st order problems (default = 1.000001).
\item replic = INTEGER: number of simulated series used to compute the IRFs (default = 1 if order = 1, and 50
otherwise).
\item simul\_seed = INTEGER or DOUBLE or (EXPRESSION): specifies a seed for the random generator so as to obtain the
same random sample at each run of the program. Otherwise a different sample is used for each run (default: seed
Going back to our good old example, suppose we are interested in printing all the various measures of moments of our variables, want to see impulse response functions for all variables, are basically happy with all default options and want to carry out simulations over a good number of periods. We would then end our .mod file with the following command:\\
\section{Executing the .mod file and interpreting results}
To see this all come to life, let's run our .mod file, which is conveniently installed by default in the Dynare ``examples'' directory (the .mod file corresponding to the stochastic model is called ExSolStoch.mod and that corresponding to the deterministic model is called ExSolDet.mod). \\
\textbf{To run a .mod file}, navigate within Matlab to the directory where the example .mod files are stored. You can do this by clicking in the ``current directory'' window of Matlab, or typing the path directly in the top white field of Matlab. Once there, all you need to do is place your cursor in the Matlab command window and type, for instance, \texttt{dynare ExSolStoch;} to execute your .mod file. \\
Running these .mod files should take at most 30 seconds. As a result, you should get two forms of output - tabular in the Matlab command window and graphical in one or more pop-up windows. Let's review these results.\\
\subsection{For stochastic models}
\textbf{The tabular results} can be summarized as follows:
\begin{enumerate}
\item\textbf{Model summary:} a count of the various variable types in your model (endogenous, jumpers, etc...).
\item\textbf{Eigenvalues} should be displayed, and you should see a confirmation of the Blanchard-Kahn conditions if you used the command \texttt{check} in your .mod file.
\item\textbf{Matrix of covariance of exogenous shocks:} this should square with the values of the shock variances and co-variances you provided in the .mod file.
\item\textbf{Policy and transition functions:} Solving the rational exectation model, $\mathbb{E}_t[f(y_{t+1},y_t,y_{t-1},u_t)]=0$ , means finding an unkown function, $y_t = g(y_{t-1},u_t)$ that could be plugged into the original model and satisfy the implied restrictions (the first order conditions). A first order approximation of this function can be written as $y_t =\bar{y}+ g_y \hat{y}_{t-1}+ g_u u_t$, with $\hat{y}_t = y_t-\bar{y}$ and $\bar{y}$ being the steadystate value of $y$, and where $g_x$ is the partial derivative of the $g$ function with respect to variable $x$. In other words, the function $g$ is a time recursive (approximated) representation of the model that can generate timeseries that will approximatively satisfy the rational expectation hypothesis contained in the original model. In Dynare, the table ``Policy and Transition function'' contains the elements of $g_y$ and $g_u$. Details on the policy and transition function can be found in Chapter \ref{ch:estadv}.
\item\textbf{Moments of simulated variables:} up to the fourth moments.
\item\textbf{Correlation of simulated variables:} these are the contemporaneous correlations, presented in a table.
\item\textbf{Autocorrelation of simulated variables:} up to the fifth lag, as specified in the options of \texttt{stoch\_simul}.
\end{enumerate}
\textbf{The graphical results}, instead, show the actual impulse response functions for each of the endogenous variables, given that they actually moved. These can be especially useful in visualizing the shape of the transition functions and the extent to which each variable is affected. \textsf{\textbf{TIP!}} If some variables do not return to their steady state, either check that you have included enough periods in your simulations, or make sure that your model is stationary, i.e. that your steady state actually exists and is stable. If not, you should detrend your variables and rewrite your model in terms of those variables.
\subsection{For deterministic models}
Automatically displayed results are much more scarce in the case of deterministic models. If you entered \texttt{steady}, you will get a list of your steady state results. Eigenvalues will also be displayed and you should receive a statement that the rank condition has been satisfied, if all goes well! (** this is despite not having entered \texttt{check}?) Finally, you will see some intermediate output: the errors at each iteration of the Newton solver used to estimate the solution to your model. \textsf{\textbf{TIP!}} You should see these errors decrease upon each iteration; if not, your model will probably not converge. If so, you may want to try to increase the periods for the transition to the new steady state (the number of simulations periods). But more often, it may be a good idea to revise your equations. Of course, although Dynare does not display a rich set of statistics and graphs corresponding to the simulated output, it does not mean that you cannot create these by hand from Matlab. To do so, you should start by looking at section \ref{sec:FindOut} of chapter \ref{ch:soladv} on finding, saving and viewing your output.
For completion's sake, and for the pleasure of seeing our work bear its fruits, here are the complete .mod files corresponding to our example for the deterministic and stochastic case. These can be found in the example folder of your Dynare installation under: \texttt{RBC\_Monop\_JFV.mod} for stochastic models and \texttt{RBC\_Monop\_Det.mod} for deterministic models.
