AR and MA together

1. AR and MA together

You've gotten to know the AR model and the MA model separately;

2. AR and MA Together: ARMA

now it's time to put them together to make one AWESOME model. Displayed is an ARMA(1,1) model. Think of it as

3. AR and MA Together: ARMA

auto-regression with correlated errors. This is natural for time series because time series data are typically correlated.

4. AR and MA Together: ARMA

The graphic shows a plot of data generated from an ARMA(1,1) model. Notice that this particular series looks like some of the actual time series we have already seen, for example, the detrended price of chicken series.

5. ACF and PACF of ARMA Models

Recall the table of ACF and PACF for the various models. You have seen the

6. ACF and PACF of ARMA Models

pure AR and pure MA. For a mixed ARMA model, both the ACF and PACF tail off. (The PACF here is not cutting off, it is just so small you can't see it after lag 5). Note that you can't tell the order of the model, just that it is ARMA. The best thing to do is to start small; that is, try an ARMA(1,1) model, and then add parameters as needed. You will see how to do this as we move along.

7. Estimation

Estimation for ARMA is basically no different than for pure models. I generated some data from this ARMA(1,1) model and then estimated the parameters using sarima. You see that the estimates are close to the actual values.

8. Let's practice!

This exercise is part of the course

ARIMA Models in R

IntermediateSkill Level

4.9+

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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

Current Exercise

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!