# Reversion to the mean The reversion to the mean is a routine that performs a variable during a time series and consists of returning to the mean value regardless of the current value of the variable.

In other words, the reversion to the mean consists in that the value that a variable takes is closer to the mean value as time passes.

## Origin of mean reversion

The reversion to the mean of a variable can occur from both high and low levels. That is to say:

• The variable takes values ​​lower than the short-term average and that in the long term it takes values ​​very close or exact to the average value.
• The variable takes values ​​higher than the short-term average and that in the long term it takes values ​​very close or exact to the average value.

## Representation of reversion to the mean

Graphical representation of reversion to the mean value of a variable over time.

Recommended articles: conditioned mean, autoregressive model, AR model, stationary stochastic process.

## Applications

The property of reversion to the mean is frequently found in the equation of the conditional mean and autoregressive models when we want to estimate the value of a variable.

## Conditional mean formula Equation of the conditioned mean.

## Formula of the conditioned mean with first-order autocorrelation AR model with mean reversion.

We can see that the reversion to the mean in the equation of the conditional mean is much more direct than in the equation of the conditional mean with autocorrelation if the coefficient G is greater than 0. If the coefficient G were equal to 0, then there would be no autocorrelation and the two equations would be identical.

For example, using the GARCH model we can estimate the reversal of current volatility towards its average level or long-term volatility.

## Volatility

• When current volatility is lower than average volatility, volatility tends to increase in the long term.
• When current volatility is higher than average volatility, volatility tends to decrease in the long term.

## Interest rates

Likewise, some equilibrium models on interest rates maintain that interest rates also enjoy this property.

The equilibrium models that support the mean reversion are the Vasicek model and the Cox, Ingresoll and Ross (CIR) model. In other words, both the Vasicek model and the CIR model share that interest rates have a long-term tendency to return to their average level.

• When interest rates are high, the economy tends to slow down and cause a decrease in the demand for capital by borrowers (investors). Then, with the passage of time (long term), interest rates fall until they reach levels close to the average level.
• When interest rates are low, the economy tends to accelerate and lead to increased demand for capital from borrowers (investors). Then, with the passage of time (long term), interest rates increase until they reach levels close to the average level.

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