R Tutorial : ARIMA Models in R

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Hi, and welcome to DataCamp's course on ARIMA modeling with R!

My name is David Stoffer. I am Professor of Statistics at the University of Pittsburgh. I am the coauthor of two texts on time series analysis. One of them, "Time Series Analysis and Its Applications: With R Examples" is the basis of this course. The text has a companion R package called astsa which stands for Applied Statistical Time Series Analysis. This package will be used throughout the course.

Now, to get started, let's explore the nature of time-series data.

Here we have the Johnson & Johnson quarterly earnings per share series. It has some common features of time series data, upward trend, seasonality in that the 2nd and 3rd quarters usually up, while the 4th quarter is usually down. In addition, there is heteroscedasticity because, as the value of the asset grows, small percent changes become large absolute changes.

The second series is the annual global temperature deviations. The data are deviations from the average temperature between 1960 and 1980. You will notice that the data have a generally positive trend, but the trend is not always positive. Unlike the Johnson and Johnson data, this series does not have a seasonal component and it is homoscedastic.

The third series is the S&P 500 weekly returns. The S&P 500 is a US stock index based on 500 large corporations. Returns are the percent change per time period. Unlike the other series, this series does not have any trend or seasonality. In fact, it seems like there are not any patterns in the series (except that once in a while, the variance is big). This is an example of a particular kind of process called noise.

Next, we will describe some models that can be used to analyze the types of time series data we have seen.

ARMA models are time series regression models. If you recall, in regression you have a dependent variable (Y), an independent variable (X), and you linearly regress Y on X. A crucial assumption is that the errors are independent, normal, and homoscedastic. In other words, the errors are white noise.

White noise is a sequence of independent normals with common variance. You will eventually see that time series models are built around white noise.

With time series, you can regress today on yesterday, and this is called an auto (or self) regression. In this case, what happens today is the dependent variable and what happened yesterday is the independent variable. As it is written, the errors are white noise.

Typically, time series data are correlated, and assuming the errors are not correlated may lead to bad forecasts. One way to overcome the problem is to use a moving average for the errors. In this example, note that the error at time t is correlated with the error at time t-1 because they both have a W_{t-1}.

Putting the two together leads to the ARMA model. In other words, the model is auto-regression with autocorrelated errors.

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