Lagged Distributed Autoregressive Model (ADR) (I)

economic-dictionary

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 passesfrom 1995 to 2018:

YearSki passes (€)TracksYearSki passes (€)Tracks
19953282007886
19964462008405
19975062009686
199855520106310
19994052011696
20003252012728
20013482013758
20026052014715
20036362015739
200464620166310
20057852017678
20068092018686
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

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