# Autoregression

Autoregression models are used to make forecasts on ex-post variables (observations that we fully know their value) at certain moments in time, normally ordered chronologically.

Autoregressive models, as their name suggests, are models that turn back on themselves. That is, the dependent variable and the explanatory variable are the same with the difference that the dependent variable will be at a later point in time (t) than the independent variable (t-1).

We say chronologically ordered because we are currently at the moment (t) of time. If we advance one period we move to (t + 1) and if we go back one period we go to (t-1).

Since we want to make a projection, the dependent variable must always be at least in a more advanced period of time than that of the independent variable. When we want to make projections using autoregression, our attention must focus on the type of variable, the frequency of its observations, and the time horizon of the projection.

## AR (p)

They are popularly known as AR (p), where p receives the 'order' label and is equivalent to the number of periods which we are going to go back to carry out the forecast of our variable. We have to take into account that the more periods we go back or the more orders we assign to the model, the more potential information will appear in our forecast.

In real life we find forecasts through autoregression in the sales projection of a company, forecast on GDP growth of a country, forecast on budget and treasury, etc.

## Estimation and forecast: result and error

The majority of the population associates the forecasts with the Ordinary Least Squares (OLS) method and the forecast error with the OLS residuals. This confusion can cause serious problems when we synthesize the information provided by the regression lines.

Difference in result:

- Estimation: The results obtained by the OLS method are calculated using observations present in the sample and have been used in the regression line.
- Forecast: Forecasts are based on a time period (t + 1) ahead of the time period of the regression observations (t). The actual forecast data for the dependent variable is not in the sample.

Difference in error:

- Estimation: the residuals (u) obtained by the OLS method are the difference between the real value of the dependent variable (Y) and the estimated value of (Y) given by the sample observations.

We remember that the subscript *Item* represents the i-th observation in the period *t*. The Y with the hat is the estimated value given the sample observations.

- Forecast: the forecast error is the difference between the future value (t + 1) of (Y), and the forecast for (Y) in the future (t + 1),. The real value of (Y) for (t + 1) does not belong to the sample.

Summary:

- The estimates and residuals belong to observations that are within the sample.
- The forecasts and their errors belong to observations that are outside the sample.

## Theoretical example of autoregression

If we want to make a forecast about the price of *ski passes* for the end of this season (t) based on the prices of last season (t-1), we can use the autoregressive model.

Our autoregressive regression would be:

This autoregressive regression belongs to the first order autoregression models or more commonly called AR. The meaning of autoregression is that the regression is done on the same variable *ski passes* but in a different period of time (t-1 and t). In the same way, it does not appear in the sample.

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