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Prediction Loss

Explore the concept of prediction loss in ARIMA time series modeling. Understand how loss functions such as mean squared error, mean absolute error, and root mean squared error help measure forecast accuracy. Learn how prediction errors arise from unknown shocks and how minimizing these loss functions is essential for optimizing model parameters and improving forecasts.

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The ultimate goal of ARIMA models is forecasting and, in particular, point forecasting. Assume that we’ve got a time series yty_t with a total of TT observations. With point forecasting, we aim to get the best possible guess for the value of yty_t, hh steps into the future. That is, we want to predict yT+hy_{T+h}.

Note: In this lesson, we use forecast and prediction interchangeably.

Prediction vs. reality

The logic of ARIMA models is that the past determines the future via some AR and MA structures. Assume that we’ve got a series yty_t that follows a known AR(22) process. To predict yT+1y_{T+1}, for example, we could simply plug the last two realizations of yty_t in the AR(22) equation:

Note: We use the hat symbol, ^, to denote parameter estimates (ϕ^1\hat \phi_1 ...