 # The Mean Reversion

Mean reversion refers to the behavior of a time series to fall when its values are above the mean and rise when they are below the mean. This is illustrated as follows: A mean-reverting time series tends to move towards its long-term mean. The model predicts that the value will stay the same if the time series is currently at its long-term mean.

i.e., $$\hat{\text{x}}_{\text{t}}=\text{x}_{\text{t}-1}$$

Substituting the above in an AR(1) model, we have:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}}$$

The mean-reverting level can be expressed as:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}}$$

$$\text{x}_{\text{t}}=\frac{\text{b}_{0}}{1-\text{b}_{1}}$$

The value of an AR(1) model will:

Remain the same when $$\text{x}_{\text{t}}=\frac{\text{b}_{0}}{1-\text{b}_{1}}$$

Increase when $$\text{x}_{\text{t}}<\frac{\text{b}_{0}}{1-\text{b}_{1}}$$

Decrease when $$\text{x}_{\text{t}}>\frac{\text{b}_{0}}{1-\text{b}_{1}}$$

#### Example: Mean Reversion

Consider the following output of an AR(1) model.

$$\begin{array}{c|c|c} {}& \textbf{Coefficients} & \textbf{Standard Error} \\ \hline \text{Intercept} & 31.4 & 4.81 \\ \hline \text{Lag 1} & -0.0006827 & 0.0001088\\ \end{array}$$

The mean-reverting level is closest to:

Solution

The mean-reverting level can be expressed as:

$$\text{x}_{\text{t}}=\frac{\text{b}_{0}}{1-\text{b}_{1}}$$

$$\text{x}_{t}=\frac{31.4}{1-(-0.0006827)}=31.39$$

It is worth noting that all covariance stationary time series will have a finite mean-reverting level. This will be discussed in more detail in future learning objectives.

## Question

Consider the following autoregressive model:

$$\text{x}_{t}=0.0407-0.5647\text{x}_{\text{t}-1}$$

The mean-reverting level is closest to:

1. 0.02601.
2. 0.03911.
3. 0.09350.

#### Solution

The mean-reverting level is given by:

$$\text{x}_{\text{t}}=\frac{\text{b}_{0}}{1-\text{b}_{1}}=\frac{0.0407}{1-(-0.5647)}=0.02601$$

LOS 5 (f) Explain mean reversion and calculate a mean-reverting level.

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