Autoregression

TL;DR

Predict future values of a time series by regressing on its past observations (lags).
Choose a number of lags and fit a regression to estimate how past values influence the next value.
Commonly applied to forecasting problems like product demand, stock prices, and weather.

Definition

Autoregression is a time series analysis technique that uses previous values of a time series to predict future values.

Explanation

Autoregression models predict a time series’ next value by using a specified number of past values (lags) as predictors. The steps described in the source are:

Identify the number of lags to include (for example, 3 lags uses the current and previous 2 months to predict the next month).
Use regression analysis to estimate the relationship between the series and its lagged values.
Form a mathematical equation from the regression coefficients to produce forecasts.

The method is applied to time-series forecasting and modeling for domains such as stock prices, product demand, and weather patterns.

Examples

Predicting monthly sales

Given monthly sales data:

Month 1: 100 units
Month 2: 120 units
Month 3: 140 units
Month 4: 160 units

Using autoregression with 3 lags, the relationship identified by regression is:

\text{Sales (Month 5)} = 100 + 20*(Sales (Month 4) - Sales (Month 3)) + 20*(Sales (Month 3) - Sales (Month 2))

Substituting the values:

\text{Sales (Month 5)} = 100 + 20*(160 - 140) + 20*(140 - 120) = 100 + 2020 + 2020 = 300 \text{ units}

Predicting stock prices

Given daily stock prices:

Day 1: $100
Day 2: $105
Day 3: $110
Day 4: $115

Using autoregression with 2 lags, the regression relationship is:

\text{Stock Price (Day 5)} = \$100 + 5*(\text{Stock Price (Day 4)} - \text{Stock Price (Day 3)}) + 5*(\text{Stock Price (Day 3)} - \text{Stock Price (Day 2)})