David Maneiro 12 19 19 Final Exam Please show all your answers in a clear format 1 Identify if the following AR models are stationary Please show all your work 20 points y t 0 8 y t 1 t a Y 0 8Y t 1 Et 1 0 8B 0 0 8B 0 8 1 0 8 B 1 25 1 This process is stationary b y t y t 1 0 5 y t 2 t Yt Yt 1 0 5Y t 2 Et Y 1 0 5B 2 B 0 1 B 0 5B 2 0 0 5B 2 B 1 0 a b c B 1 2 b square root b 2 4ac 2a B 1 2 1 square root 1 2 4 0 5 1 1 square root 1 2 2 0 5 1 1 73 1 2 73 1 1 1 73 1 0 73 1 This process is not stationary 2 Using the variable that was assigned to you please answer the following questions 30 points a Fit an Ar 1 model and plot forecasts for the next 3 periods b Fit an Ar 2 model and plot forecasts for the next 3 periods c Which model do you think produced the most reliable forecast values Please show your work Fit 1 MSE 0 01 1 27 2 21 3 1 16 RMSE 1 07 Fit 2 MSE 1 0 5 0 13 3 0 54 RMSE 0 73 AR2 is more reliable because it gives us a lower RMSE d Are the estimated model stationary B1 1 7519 b2 4ac 2a 1 7519 3 069 3 00076 1 1 7519 3 069 3 0076 2 0 7519 1 7519 0 0614 1 5038 AR1 1 33 1 B1 is stationary B2 1 7519 0 0614 1 5038 AR2 1 0002 1 B2 is stationary 3 Using one of the exponential smoothing models forecast three period for the assigned variable You have to explain how you decided on which exponential smoothing to use 20 points Holt model SES model Looking at the RSME of the of the Holts and SES smoothing models we see that the holt model gives us a lower RMSE so that will be the exponential smoothing model that I will be using to forecast three periods for the CPI Forecasted periods for CPI 4 Now if you have to choose between the AR model and the exponential smoothing model which model do you think is more reliable to forecast your assigned variable 20 points Looking at the RMSE between the SES Holts AR 1 and AR 2 models I would have to say that the AR 2 model is the most reliable with the Holt behind in second and the SES and AR 1 in third and fourth respectively in terms of reliability 5 Please show if the assigned variable has a trend and seasonality 10 points Clearly CPI has an upward trend however looking at the graph it does not seem to have seasonality R studio coding library ggplot2 library forecast library tseries library TSA data read csv file choose header TRUE cpi ts data CPI frequency 4 start c 1966 1 fit arima cpi order c 1 0 0 seasonal c 0 0 0 fit plot fit res residuals fit plot res checkresiduals fit forecpi forecast cpi model fit h 3 plot cpi forecpi fit2 arima cpi order c 2 0 0 seasonal c 0 0 0 fit2 plot fit2 res2 residuals fit2 plot res2 checkresiduals fit2 forecpi2 forecast cpi model fit2 h 3 forecpi2 cpi2 cpi 1 210 fit3 arima cpi2 order c 1 0 0 seasonal c 0 0 0 fit3 plot fit3 res3 residuals fit3 plot res3 checkresiduals fit3 forecpi3 forecast cpi2 model fit3 h 3 plot cpi2 forecpi3 fit4 arima cpi2 order c 2 0 0 seasonal c 0 0 0 fit4 plot fit4 res4 residuals fit4 plot res4 checkresiduals fit4 forecpi4 forecast cpi2 model fit4 h 3 plot cpi2 forecpi4 fc1 holt cpi h 3 plot fc1 fc1 summary fc1 e tsCV cpi holt h 3 e2 tsCV cpi hw h 3 e e2 fc2 ses cpi h 3 round accuracy fc2 3 summary fc2 plot fc2 fc2 autoplot cpi xlab DATE ylab cpi