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).
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.
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|
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