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(This article was first published on <strong><a href="http://robjhyndman.com/hyndsight/simulating-from-a-specified-seasonal-arima-model/"> R – Hyndsight</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p>From my email today</p>
<blockquote><p>You use an illustration of a seasonal arima model:</p>
<p>ARIMA(1,1,1)(1,1,1)4</p>
<p>I would like to simulate data from this process then fit a model… but I am unable to find any information as to how this can be conducted… if I set phi1, Phi1, theta1, and Theta1 it would be reassuring that for large n the parameters returned by <code>Arima(foo,order=c(1,1,1),seasonal=c(1,1,1))</code> are in agreement…</p>
</blockquote>
<h4>My answer:</h4>
<p>Unfortunately <code>arima.sim()</code> won’t handle seasonal ARIMA models. I wrote <code>simulate.Arima()</code> to handle them, but it is designed to simulate from a fitted model rather than a specified model. However, you can use the following code to do it. It first “estimates” an ARIMA model with specified coefficients. Then simulates from it.</p>
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<pre><span>library</span><span>(</span>forecast<span>)</span>
model <span>&lt;-</span> Arima<span>(</span><span>ts</span><span>(</span><span>rnorm</span><span>(</span><span>100</span><span>)</span>,freq<span>=</span><span>4</span><span>)</span>, <span>order</span><span>=</span><span>c</span><span>(</span><span>1</span>,<span>1</span>,<span>1</span><span>)</span>, seasonal<span>=</span><span>c</span><span>(</span><span>1</span>,<span>1</span>,<span>1</span><span>)</span>,
             fixed<span>=</span><span>c</span><span>(</span>phi<span>=</span><span>0.5</span>, theta<span>=-</span><span>0.4</span>, Phi<span>=</span><span>0.3</span>, Theta<span>=-</span><span>0.2</span><span>)</span><span>)</span>
foo <span>&lt;-</span> <span>simulate</span><span>(</span>model, nsim<span>=</span><span>1000</span><span>)</span>
fit <span>&lt;-</span> Arima<span>(</span>foo, <span>order</span><span>=</span><span>c</span><span>(</span><span>1</span>,<span>1</span>,<span>1</span><span>)</span>, seasonal<span>=</span><span>c</span><span>(</span><span>1</span>,<span>1</span>,<span>1</span><span>)</span><span>)</span></pre>
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