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  • Unconstrained matching
  • Constrained matching:
    • Match the value of the initial raterate 
      LaTeX Math Inline
      body--uriencoded--q%5e*(t=0) = q_0
    • Match the value of the current rate
      LaTeX Math Inline
      body--uriencoded--q%5e*(t=t_N) = q_N
    • Match the value of the current cumulative
      LaTeX Math Inline
      body--uriencoded--Q%5e*(t=t_N) = Q_N
    • Match the value of the current rate and cumulative
      LaTeX Math Inline
      body--uriencoded--q%5e*(t=t_N) = q_N
      ,
      LaTeX Math Inline
      body--uriencoded--Q%5e*(t=t_N) = Q_N


To ensure the smooth transition from historical data

LaTeX Math Inline
body[(t_1,q_1)... (t_N, q_N)]
 to the production forecasts in future time moments
LaTeX Math Inline
body--uriencoded--[(t_%7BN+1%7D,q_%7BN+1%7D), ...]
one may wish to constrain the model by firm matching the production at the last historical moment
LaTeX Math Inline
body(t_N, q_N)
 which leads to the following form of Arp's model:

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