# Lagged Distributed Autoregressive Model (ADR) (I)

The Lagged Distributed Autoregressive (ADR) model, from English *Autoregressive Distributed Lag Model*(ADL), is a regression involving a new lagged independent variable in addition to the lagged dependent variable.

In other words, the ADR model is an extension of the p-order autoregressive model, AR (p), which includes another independent variable in a period of time prior to the period of the dependent variable.

The ADR model is expressed as ADR (p, q), where:

p = are the lagged periods of the dependent variable (Y).

q = are the lagged periods of the additional independent variable (X).

### Mathematically

Model AR (p):

New additional independent variable (X):

ADR model (p, q):

The ADR model is called *autoregressive *because the regression includes lagged values during *p *periods of the dependent variable as regressors. *Distributed lagging *because the regression also incorporates other values lagged during *what* periods of an additional independent variable.

We define the error term (ut) and assume:

This assumption implies that other lagged values of Y and X do not belong to the ADR model. That is, the all lagged values are between Yt-p and Xt-q.

We recommend reading the article: natural logarithms, AR.

## Practical example

We suppose that we want to study the price of *ski passes* for this season 2019 (t) depending on the prices of the passes and the number of black slopes open from the previous season (t-1). So, instead of using the AR (p) model, we can apply the ADR (p, q) model since it incorporates both independent variables: *ski pass*-1y *pistast*-1.

The model would be:

We have the prices of the *ski passes*from 1995 to 2018:

Year | Ski passes (€) | Tracks | Year | Ski passes (€) | Tracks |

1995 | 32 | 8 | 2007 | 88 | 6 |

1996 | 44 | 6 | 2008 | 40 | 5 |

1997 | 50 | 6 | 2009 | 68 | 6 |

1998 | 55 | 5 | 2010 | 63 | 10 |

1999 | 40 | 5 | 2011 | 69 | 6 |

2000 | 32 | 5 | 2012 | 72 | 8 |

2001 | 34 | 8 | 2013 | 75 | 8 |

2002 | 60 | 5 | 2014 | 71 | 5 |

2003 | 63 | 6 | 2015 | 73 | 9 |

2004 | 64 | 6 | 2016 | 63 | 10 |

2005 | 78 | 5 | 2017 | 67 | 8 |

2006 | 80 | 9 | 2018 | 68 | 6 |

2019 | ? |

We only go back one period, so:

p = are the lagged periods of the dependent variable (*ski pass*) = 1

q = are the lagged periods of the additional independent variable (*pistast*)= 1

ADR (p, q) = ADR

We could incorporate more variables relevant to the model and increase lag periods in each variable up to ADR (p, q).

ADR solved example