# 18 AR(1) Model

A univariate time series is a sequence of measurements of the same variable collected over time. Most often, the measurements are made at regular time intervals.

One defining characteristic of a time series is that it is a list of observations where the **ordering matters**. Ordering is very important because there is dependency and changing the order could change the meaning of the data.

Some important questions to first consider when first looking at a time series are:

- Is there a
**trend**, meaning that, on average, the measurements tend to increase (or decrease) over time? - Is there
**seasonality**, meaning that there is a regularly repeating pattern of highs and lows related to calendar time such as seasons, quarters, months, days of the week, and so on? - Are there
**outliers**? In regression, outliers are far away from your line. With time series data, your outliers are far away from your other data. - Is there a
**long-run cycle**or period unrelated to seasonality factors? - Is there
**constant variance**over time, or is the variance non-constant? - Are there any
**abrupt changes**to either the level of the series or the variance?

^{2}

The following plot is a time series plot of the **annual number of earthquakes in the world with seismic magnitude over 7.0**, for 99 consecutive years.

Some features of the time series plot^{3}:

- There is no consistent trend (upward or downward) over the entire time span. The series appears to slowly wander up and down. The horizontal line drawn at quakes = 20.2 indicates the mean of the series. Notice that the series tends to stay on the same side of the mean (above or below) for a while and then wanders to the other side.
- Almost by definition, there is no seasonality as the data are annual data.
- There are no obvious outliers.
- It is difficult to judge whether the variance is constant or not.

One of the simplest *ARIMA* type models is a model in which we use a linear model to predict the value at the present time using the value at the previous time. This is called an **AR(1)** model, standing for **autoregressive model of order 1**. The order of the model indicates how many previous times we use to predict the present time.

A start in evaluating whether an **AR(1)** might work is to plot values of the series against lag 1 values of the series: \(y_t\) vs $y_{t_{1}}.

Although it’s only a moderately strong relationship, there is a positive linear association so an AR(1) model might be a useful model.

By a time series plot, we simply mean that the variable is plotted against time.↩︎