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Applying both seasonal and regular differencing to a time series ensures that the differenced series will be stationary.

After fitting a model to time series data, residuals are obtained. If we run a Shapiro Wilk test, and get a p-value of 0.84. What conclusion would you make?

If a time series follows an AR(1) process, the autocorrelation function

Which of the following properties are necessary for weak stationarity in time series? Choose all correct answers.

Select the correct statements. Choose all correct answers.  

Which of the following are advantages of using Maximum Likelihood Estimation (MLE) in time series analysis? Choose all correct answers.

If the moving average width in a trend estimation model is increased, the estimated trend becomes smoother.

For a white noise process, which of the following are true? Choose all correct answers.

Which of the following are techniques used to estimate the parameters of an ARMA model? Choose all correct answers.

ARIMA Modeling and Forecasting 2a. Split the data in Training and Testing sets. Leave the last 8 observations for testing, the rest will be your training set. You will apply the same trend-seasonality model from Question 1d on the Training set and forecast the corresponding data for the testing set. Calculate the MAPE and interpret it. How does it compare to the MAPE in 1(d)? Why is it different? Hints:  – Keep in mind that modeling factors may require extra steps on the data preparation.  – You can use predict, or predict.gam for your predictions.  2b. Fit an ARMA model using the residuals from the model in Question 2a. Find the order of the ARMA model using a max order 6 for p and q, and 1 for d.  Use AICc as the criterion for the order selection. What are the selected orders? Use the forecast from the model you find and add them to the forecast values in 2a. Comment on what would be the logic of this procedure? 2c. Fit an SARIMA model to the training dataset using ARIMA orders (10,1,8) and seasonal orders (1,0,1). Comment on the model goodness of fit. 2d. Use the model from Question 2c to forecast the next eight time points (testing dataset) using the **8 lags ahead approach**. Overlay the observed versus predicted values for both series, including 95% confidence intervals. Calculate the MAPE of the prediction and comment on the prediction performance of the model. What approach would you use for this data? Why?