STAT 443: Time Series and Forecasting Lab 4: SARIMA Processes We have seen both
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STAT 443: Time Series and Forecasting Lab 4: SARIMA Processes We have seen both autoregressive (AR) and moving average (MA) models, as well as hybrids of these known as autoregressive moving average (ARMA) models. While these are quite versatile models, recall that they are used to model series that are stationary 1. Without using any mathematical notation, describe in words what it means for a time series to be stationary 2. Consider a realization of a process given by the values (3.89, 8.04, 10.26, 10.72, 10.69, 12.50, 16.43, 20.15, 22.38, 22.45, 21.91, 24.06, 28.03, 32.35, 34.47, 3447, 3498, 36.36, 39.86, 43.57) Enter these data into R, plot as a time series, and comment on whether the series appears to satisfy the requirements of stationarity 3. One common way of removing a trend is to difference the data, where instead of looking at the time series (re) we look at (w) with (Note this series will have one less term than the original.) Using the data above, determine and plot the differenced time series (r). Comment on the resulting plot and the ado(n. A useful function here is diff (x, lag-1, difference-1), which returns suitably lagged and iterated differences. Use R help to learn about options 1ag and ditferences 4. In order to remove a seasonal effect, we could difference over the seasonal period. For instance, take our de-trended data (ve and difference again but at a lag equal to the seasonal period, s. The new series will be where s is the period of the seasonal effect. Note the series will again become shorter, this time with s fewer terms. Choosing the appropriate value of s here, apply seasonal differencing to the series from part 3 and plot the resulting series. Plot the ad of V Does your le resemble white noise? 5. Suggest which type of model from the SARIMA family you would use for the original data. 6. Use the arima function to fit a SARIMA model to the original (undifferenced) time series Provide the estimates of any parameters.Explanation / Answer
Q1. A stationary time series is one whose statistical properties such as mean, variance, autocorrelation, etc. are all constant over time. Most statistical forecasting methods are based on the assumption that the time series can be rendered approximately stationary through the use of mathematical transformations. A stationary series is relatively easy to predict: you simply predict that its statistical properties will be the same in the future as they have been in the past.
A stationary time series helps us obtain meaningful sample statistics such as means, variances, and correlations with other variables. Such statistics are useful as descriptors of future behavior only if the series is stationary. For example, if the series is consistently increasing over time, the sample mean and variance will grow with the size of the sample, and they will always underestimate the mean and variance in future periods.
A stationary time series is an important part of the process of fitting an ARIMA model.
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