Quick questions on Time Series and Forecasting - TCE Mathematics Applications (Tasmania)

A moving average replaces each value with the average of itself and its neighbours, which smooths out short-term fluctuations and reveals the trend. The number of points averaged is the order. For an odd order (e.g. 3 or 5), the smoothed value lines up directly with a data point.

A seasonal index measures how a particular season compares with the average. An index of $1.20$ means that season runs $20\%$ above the average; $0.85$ means $15\%$ below. For data with $n$ seasons, the indices must add to $n$ (for example, four quarterly indices sum to $4$, and average to $1$).

Once you have a trend line (often a least-squares line fitted to time $t$ on the horizontal axis), substitute the future time value to forecast. As with regression, forecasting far beyond the data is extrapolation and grows less reliable the further out you go.

To find seasonal indices yourself, average each season's values, divide each seasonal average by the overall average of all the data, and the results are the indices. They must sum to the number of seasons (4 for quarterly, 12 for monthly), so if they do not, scale them: multiply each by the number of seasons divided by their current sum. This forces the average index to 1, which is the property that makes deseasonalising and reseasonalising consistent.