Exploration of Autocorrelation Concepts

Chapter 2: Lesson 3

Learning Outcomes

Explain the theoretical implications of autocorrelation for the estimation of time series statistics
  • Explain how positive autocorrelation leads to underestimation of variance in short time series
  • Explain how negative autocorrelation can improve efficiency of sample mean estimate
Interpret correlograms to identify significant lags, correlations, trends, and seasonality
  • Create a correlogram
  • Interpret a correlogram
  • Define a sampling distribution
  • State the sampling distribution of rk
  • Explain the concept of a confidence interval
  • Conduct a single hypothesis test using a correlogram
  • Describe the problems associated with multiple hypothesis testing in a correlogram
  • Differentiate statistical and practical significance
  • Diagnose non-stationarity using a correlogram

Preparation

  • Read Sections 2.3-2.5

Learning Journal Exchange (10 min)

  • Review another student’s journal
  • What would you add to your learning journal after reading your partner’s?
  • What would you recommend your partner add to their learning journal?
  • Sign the Learning Journal review sheet for your peer

Correlograms (10 min)

In the previous lesson, we used the following time series as an example. Here are the values in that time series:

x <- c( 4.4, 4.2, 4.2, 4, 4.4, 4.7, 4.9, 5.3, 5.4, 5.5 )
  • The table below gives the sample autocorrelation function, acf, for this data set. You may recognize some of these values from the previous lesson.
0 1 2 3 4 5 6 7 8 9
1 0.763 0.448 0.074 -0.237 -0.419 -0.47 -0.344 -0.226 -0.089
Check Your Understanding
  • Use the acf values to sketch the correlogram for these data in your Learning Journal. The figure below can help you begin.

  • Are any of the autocorrelations statistically significant? If so, which one(s)?

Small Group Activity: BYU-Idaho On-Campus Enrollment (25 min)

The official number of on-campus BYU-Idaho students each semester is given in the file byui_enrollment.csv.

Check Your Understanding

Do the following:

  • Create a tsibble with the BYU-Idaho enrollment data. (Hint: There are three semesters in a year, so treat the enrollments as observations taken every four months in January, May, and September.)
  • Plot the decomposition of this time series.
  • Describe the trend.
  • Describe the seasonal component.
  • Is there evidence of seasonal variation? If so, propose an explanation for the seasonal variation.
  • Create the correlogram for these data.
    • What do you observe?
    • Does the correlogram support the statement you made about the seasonal component?
  • Is there evidence of autocorrelation in the data after removing the trend and seasonal variation?

Homework Preview (5 min)

  • Review upcoming homework assignment
  • Clarify questions

Homework

Download Homework

Correlograms

BYU-Idaho Enrollment