Fitting an AR(1) model

Recall that you use the ACF and PACF pair to help identify the orders \(p\) and \(q\) of an ARMA model. The following table is a summary of the results:

	AR(\(p\))	MA(\(q\))	ARMA(\(p,q\))
ACF	Tails off	Cuts off after lag \(q\)	Tails off
PACF	Cuts off after lag \(p\)	Tails off	Tails off

In this exercise, you will generate data from the AR(1) model, $$X_t = .9 X_{t-1} + W_t,$$ look at the simulated data and the sample ACF and PACF pair to determine the order. Then, you will fit the model and compare the estimated parameters to the true parameters.

Throughout this course, you will be using sarima() from the astsa package to easily fit models to data. The command produces a residual diagnostic graphic that can be ignored until diagnostics is discussed later in the chapter.

This is a part of the course

“ARIMA Models in R”

Exercise instructions

The package astsa is preloaded.
Use the prewritten arima.sim() command to generate 100 observations from an AR(1) model with AR parameter .9. Save this to x.
Plot the generated data using plot().
Plot the sample ACF and PACF pairs using the acf2() command from the astsa package.
Use sarima() from astsa to fit an AR(1) to the previously generated data. Examine the t-table and compare the estimates to the true values. For example, if the time series is in x, to fit an AR(1) to the data, use sarima(x, p = 1, d = 0, q = 0) or simply sarima(x, 1, 0, 0).

Hands-on interactive exercise

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

# Generate 100 observations from the AR(1) model
x <- arima.sim(model = list(order = c(1, 0, 0), ar = .9), n = 100) 

# Plot the generated data 


# Plot the sample P/ACF pair


# Fit an AR(1) to the data and examine the t-table

Edit and Run Code

This exercise is part of the course

ARIMA Models in R

IntermediateSkill Level

4.5+

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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 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!