Identify the p and q values for the following ARMA models written in the characteristic equation.
\((1−\frac{1}{2}B)(1−\frac{1}{3}B)x_t = w_t\)
\((1−\frac{1}{2}B)x_t = (1−\frac{1}{3}B)w_t\)
Express the following ARMA models with the characteristic equation.
\(x_t= \frac{1}{2} x_{t-1} + w_t + \frac{1}{4} w_{t-1}\)
\(x_t= w_t - \frac{3}{4} w_{t-1} + \frac{1}{8} w_{t-2}\)
Include code for this question. Note the simulations should be conducted several times to limit the Monte Carlo variation in your answers.
Simulate a MA(1) process with \(\beta = .5\) and estimate the lag 1 and lag 2 correlation (you can use the arima.sim function). That is estimate \(\rho(x_t, x_{t+1})\) and \(\rho(x_t, x_{t+1})\).
How do the results match with your expectations?
Simulate a ARMA(1,1) process with \(\alpha = .7\) and \(\beta = .5\) and estimate the lag 1 and lag 2 correlation (you can use the arima.sim function). That is estimate \(\rho(x_t, x_{t+1})\) and \(\rho(x_t, x_{t+1})\).
How do the results match with your expectations?
Evaluate the code below and describe the results.
library(readr)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(forecast)
bakery.sales <- read_csv('http://math.montana.edu/ahoegh/teaching/timeseries/data/BreadBasket.csv')## Parsed with column specification:
## cols(
## Date = col_date(format = ""),
## Time = col_time(format = ""),
## Transaction = col_integer(),
## Item = col_character()
## )pastry.count <- bakery.sales %>% filter(Item %in% c('Pastry','Scandinavian','Medialuna','Muffin','Scone')) %>% group_by(Date) %>% tally() %>% rename(num.pastry = n) %>% select(num.pastry) %>% pull() %>% ts(frequency =7)ggtsdisplay(pastry.count)
auto.arima(pastry.count, max.d = 0, seasonal = F)ggtsdisplay(pastry.count)
auto.arima(pastry.count, max.d = 0, max.D = 0)