David Maneiro12/19/19Final ExamPlease show all your answers in a clear format.1. Identify if the following AR models are stationary. Please show all your work. (20 points)a.yt=0.8 yt−1+εtY+ -0.8Y t-1 = Et1-0.8B=0-0.8B/0.8 = -1/-0.8 B = 1.25 > 1 This process is stationary.b.yt= yt−1−0.5 yt −2+ εtYt – Yt-1 + 0.5Y t-2 = EtY + (1 + 0.5B^2 – B) = 01-B+0.5B^2 = 00.5B^2 – B + 1 = 0(a) (b) (c)B^1/2 = -b + square root ( b^2 – 4ac)/2aB^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 > 11- 1.73/1 = -0.73 < 1This process is not stationary2. Using the variable that was assigned to you, please answer the following questions: (30points)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= 1.16RMSE= 1.07Fit 2MSE= (-1+-0.5+-0.13)/3= -0.54RMSE= 0.73AR2 is more reliable because it gives us a lower RMSEd. 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 modelSES modelLooking 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 willbe using to forecast three-periods for the CPIForecasted periods for CPI4. 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 tosay that the AR(2) model is the most reliable, with the Holt behind in second and the SESand 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 codinglibrary(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))fitplot(fit)res=residuals(fit)plot(res)checkresiduals(fit)forecpi=forecast(cpi, model = fit, h=3)plot(cpi)forecpifit2=arima(cpi, order = c(2,0,0), seasonal = c(0,0,0))fit2plot(fit2)res2=residuals(fit2)plot(res2)checkresiduals(fit2)forecpi2=forecast(cpi, model = fit2, h=3)forecpi2cpi2=cpi[1:210]fit3=arima(cpi2, order = c(1,0,0), seasonal = c(0,0,0))fit3plot(fit3)res3=residuals(fit3)plot(res3)checkresiduals(fit3)forecpi3=forecast(cpi2, model = fit3, h=3)plot(cpi2)forecpi3fit4=arima(cpi2, order = c(2,0,0), seasonal = c(0,0,0))fit4plot(fit4)res4=residuals(fit4)plot(res4)checkresiduals(fit4)forecpi4=forecast(cpi2, model = fit4, h=3)plot(cpi2)forecpi4fc1=holt(cpi, h=3)plot(fc1)fc1summary(fc1)e=tsCV(cpi, holt, h=3)e2=tsCV(cpi, hw,h=3)ee2fc2=ses(cpi, h=3)round(accuracy(fc2),3)summary(fc2)plot(fc2)fc2autoplot(cpi)+xlab("DATE") +