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.

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):

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 passesfrom 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

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

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