When estimating any model on a sample of data, we will inevitably have to deal with uncertainty. Consider an example, when we want to estimate the average height of a person in the room. We could take heights of all the people in the room, then take average and we would get our answer. But what would happen with that average if another person comes in the room? We would need to do additional measures and re-estimate the average, and inevitably it will be different from the one we had before. This example demonstrates one of the classical sources of uncertainty - the one caused by estimation on a sample of data.
Furthermore, we might be interested in predicting the weight of a person based on their height. The two variables will be related, but would not have a functional relation: with the increase of height we expect that a person will weigh more, but this only holds on average. So, based on a sample of data, we could estimate the relation between the two variables and then having a height of a person, we could predict their expected weight. Their individual weight will inevitably vary from one person to another. This is the second source of uncertainty, appearing because of the individual discrepancies from one person to another.
Finally, the model of weight from height could be wrong for different reasons. For example, there might be plenty of other factors that would impact the weight of person that we have not taken into account. In fact, we never know the true model (see Section 1.1.1), so this is the third source of uncertainty, the one around the model form.
These three sources of uncertainty have been summarised for the first time in Chatfield (1996). Whenever we need to construct any type of model, we will deal with:
- Uncertainty about the data, e.g. the error term \(\epsilon_t\) (see Section 6);
- Uncertainty about estimates of parameters (see Section 12);
- Uncertainty about the model form (see Section 17).
In these lecture notes we will discuss all of them, slowly moving from (1) to (3), introducing more advanced techniques for model building.
• Chatfield, C., 1996. Model uncertainty and forecast accuracy. Journal of Forecasting. 15, 495–508. https://doi.org/10.1002/(SICI)1099-131X(199612)15:7<495::AID-FOR640>3.3.CO;2-F