You just applied a linear regression model to time series data and saw it did not work. The biggest problem was not a failure in fitting a linear model to the trend. For this well-behaved time series, the average formed a linear plot over time. Where was the problem?
The problem was in seasonal fluctuations. The seasonal fluctuations were one year in length and then repeated. Most of the data points existed above and below the fitted line, instead of on it or near it. As we saw, the ability to make a point estimate prediction was poor. There is an old adage that says even a broken clock is correct twice a day. This is a good analogy for analyzing seasonal time series data with linear regression. The fitted linear line will be a good predictor twice every cycle. You will need to do something about the seasonal fluctuations in order to make better forecasts; otherwise, they will simply be straight lines with no account of the seasonality.
With...