Detrending

Some nonstationary series can be modeled as the sum of a deterministic trend and a stationary stochastic process. That is, you can write the series y_t as

where $${\epsilon }_t$$ is a stationary stochastic process with mean zero.

The deterministic trend, μ_t, can have multiple components, such as nonseasonal and seasonal components. You can detrend (or decompose) the data to identify and estimate its various components. The detrending process proceeds as follows:

Several techniques are available for estimating the trend component. You can estimate it parametrically using least squares, nonparametrically using filters (moving averages), or a combination of both.

Detrending yields estimates of all trend and stochastic components, which might be desirable. However, estimating trend components can require making additional assumptions, performing extra steps, and estimating additional parameters.

Differencing

Differencing is an alternative transformation for removing a mean trend from a nonstationary series. This approach is advocated in the Box-Jenkins approach to model specification [1]. According to this methodology, the first step to build models is differencing your data until it looks stationary. Differencing is appropriate for removing stochastic trends (e.g., random walks).

Define the first difference as

where Δ is called the differencing operator. In lag operator notation, where $$L^i{y}_t=y_{t-i},$$

You can create lag operator polynomial objects using LagOp.

Similarly, define the second difference as

Like taking derivatives, taking a first difference makes a linear trend constant, taking a second difference makes a quadratic trend constant, and so on for higher-degree polynomials. Many complex stochastic trends can also be eliminated by taking relatively low-order differences. Taking D differences makes a process with D unit roots stationary.

For series with seasonal periodicity, seasonal differencing can address seasonal unit roots. For data with periodicity s (e.g., quarterly data have s = 4 and monthly data have s = 12), the seasonal differencing operator is defined as

Using a differencing transformation eliminates the intermediate estimation steps required for detrending. However, this means you can’t obtain separate estimates of the trend and stochastic components.

Log Transformations

For a series with exponential growth and variance that grows with the level of the series, a log transformation can help linearize and stabilize the series. If you have negative values in your time series, you should add a constant large enough to make all observations greater than zero before taking the log transformation.

In some application areas, working with differenced, logged series is the norm. For example, the first differences of a logged time series,

are approximately the rates of change of the series.

Prices, Returns, and Compounding

The rates of change of a price series are called returns. Whereas price series do not typically fluctuate around a constant level, the returns series often looks stationary. Thus, returns series are typically used instead of price series in many applications.

Denote successive price observations made at times t and t + 1 as y_t and y_t+1, respectively. The continuously compounded returns series is the transformed series

This is the first difference of the log price series, and is sometimes called the log return.

An alternative transformation for price series is simple returns,

For series with relatively high frequency (e.g., daily or weekly observations), the difference between the two transformations is small. Econometrics Toolbox has price2ret for converting price series to returns series (with either continuous or simple compounding), and ret2price for the inverse operation.