title: Explaining the forecasts from an ARIMA model
body: <p>I am trying to explain to myself the forecasting result from applying an ARIMA model to a time-series dataset. The data is from the M1-Competition, the series is MNB65. I am trying to fit the data to an ARIMA(1,0,0) model and get the forecasts. I am using R. Here are some output snippets:</p>
  
  <pre><code>&gt; arima(x, order = c(1,0,0))
  Series: x 
  ARIMA(1,0,0) with non-zero mean 
  Call: arima(x = x, order = c(1, 0, 0)) 
  Coefficients:
           ar1  intercept
        0.9421  12260.298
  s.e.  0.0474    202.717
  
  &gt; predict(arima(x, order = c(1,0,0)), n.ahead=12)
  $pred
  Time Series:
  Start = 53 
  End = 64 
  Frequency = 1 
  [1] 11757.39 11786.50 11813.92 11839.75 11864.09 11887.02 11908.62 11928.97 11948.15 11966.21 11983.23 11999.27
  </code></pre>
  
  <p>I have a few questions:</p>
  
  <p>(1) How do I explain that although the dataset shows a clear downward trend, the forecast from this model trends upward? This also happens for ARIMA(2,0,0), which is the best ARIMA fit for the data using <code>auto.arima</code> (forecast package) and for an ARIMA(1,0,1) model.</p>
  
  <p>(2) The intercept value for the ARIMA(1,0,0) model is 12260.298. Shouldn't the intercept satisfy the equation: <code>C = mean * (1 - sum(AR coeffs))</code>, in which case, the value should be <code>715.52</code>. I must be missing something basic here.</p>
  
  <p>(3) This is clearly a series with non-stationary mean. Why is an AR(2) model still selected as the best model by <code>auto.arima</code>? Could there be an intuitive explanation?</p>
  
  <p>Thanks.</p>