Differencing

As seen in the video, when a time series is trend stationary, it will have stationary behavior around a trend. A simple example is \(Y_t = \alpha + \beta t + X_t\) where \(X_t\) is stationary.

A different type of model for trend is random walk, which has the form \(X_t = X_{t-1} + W_t\), where \(W_t\) is white noise. It is called a random walk because at time \(t\) the process is where it was at time \(t-1\) plus a completely random movement. For a random walk with drift, a constant is added to the model and will cause the random walk to drift in the direction (positive or negative) of the drift.

We simulated and plotted data from these models. Note the difference in the behavior of the two models.

In both cases, simple differencing can remove the trend and coerce the data to stationarity. Differencing looks at the difference between the value of a time series at a certain point in time and its preceding value. That is, \(X_t - X_{t-1}\) is computed.

To check that it works, you will difference each generated time series and plot the detrended series. If a time series is in x, then diff(x) will have the detrended series obtained by differencing the data. To plot the detrended series, simply use plot(diff(x)).

This exercise is part of the course

ARIMA Models in R

Exercise instructions

In one line, difference and plot the detrended trend stationary data in y by nesting a call to diff() within a call to plot(). Does the result look stationary?
Do the same for x. Does the result look stationary?

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Plot detrended y (trend stationary)


# Plot detrended x (random walk)

Edit and Run Code

This exercise is part of the course

ARIMA Models in R

IntermediateSkill Level

5.0+

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You will investigate the nature of time series data and learn the basics of ARMA models that can explain the behavior of such data. You will learn the basic R commands needed to help set up raw time series data to a form that can be analyzed using ARMA models.

Exercise 1: First things first Exercise 2: Data play Exercise 3: Elements of time series Exercise 4: Stationarity and nonstationarity Exercise 5: Differencing

Current Exercise

Exercise 6: Detrending data Exercise 7: Dealing with trend and heteroscedasticity Exercise 8: Stationary time series: ARMA Exercise 9: Simulating ARMA models

You will discover the wonderful world of ARMA models and how to fit these models to time series data. You will learn how to identify a model, how to choose the correct model, and how to verify a model once you fit it to data. You will learn how to use R time series commands from the stats and astsa packages.

Exercise 1: AR and MA models Exercise 2: Fitting an AR(1) model Exercise 3: Fitting an AR(2) model Exercise 4: Fitting an MA(1) model Exercise 5: AR and MA together Exercise 6: Fitting an ARMA model Exercise 7: Identify an ARMA model Exercise 8: Model choice and residual analysis Exercise 9: Model choice - I Exercise 10: Model choice - II Exercise 11: Residual analysis - I Exercise 12: Residual analysis - II Exercise 13: ARMA get in

Now that you know how to fit ARMA models to stationary time series, you will learn about integrated ARMA (ARIMA) models for nonstationary time series. You will fit the models to real data using R time series commands from the stats and astsa packages.

Exercise 1: ARIMA - integrated ARMA Exercise 2: ARIMA - plug and play Exercise 3: Simulated ARIMA Exercise 4: Global warming Exercise 5: ARIMA diagnostics Exercise 6: Diagnostics - simulated overfitting Exercise 7: Diagnostics - global temperatures Exercise 8: Forecasting ARIMA Exercise 9: Forecasting simulated ARIMA Exercise 10: Forecasting global temperatures

You will learn how to fit and forecast seasonal time series data using seasonal ARIMA models. This is accomplished using what you learned in the previous chapters and by learning how to extend the R time series commands available in the stats and astsa packages.

Exercise 1: Pure seasonal models Exercise 2: P/ACF of pure seasonal models Exercise 3: Fit a pure seasonal model Exercise 4: Mixed seasonal models Exercise 5: Fit a mixed seasonal model Exercise 6: Data analysis - unemployment I Exercise 7: Data analysis - unemployment II Exercise 8: Data analysis - commodity prices Exercise 9: Data analysis - birth rate Exercise 10: Forecasting seasonal ARIMA Exercise 11: Forecasting monthly unemployment Exercise 12: How hard is it to forecast commodity prices?Exercise 13: Congratulations!