Here are some results that show the consequences of extending the Michigan January low temperature analysis from 2019 to 2021.
y <- read.table(file="ann_arbor_weather.csv",header=1)
arma2021 <- arima(y$Low, order=c(1,0,1))
arma2021
##
## Call:
## arima(x = y$Low, order = c(1, 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## -0.5611 0.5921 -2.7394
## s.e. 0.7481 0.7302 0.6944
##
## sigma^2 estimated as 56.12: log likelihood = -415.36, aic = 838.71
arma2019 <- arima(y$Low[y$Year<=2019], order=c(1,0,1))
arma2019
##
## Call:
## arima(x = y$Low[y$Year <= 2019], order = c(1, 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## 0.7852 -0.7414 -2.9680
## s.e. 0.3184 0.3429 0.8115
##
## sigma^2 estimated as 54.52: log likelihood = -406.77, aic = 821.55
These results may be a bit surprising. Why? Can you interpret them based on what we saw in class: plotting the data, writing the equation for the ARMA(1,1) model, and understanding that arima()
carries out maximum likelihood estimation?