Chapter 1 discussion question

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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?

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