**4. Pretesting**

Let us consider the situation where *tj* is used as a diagnostic in more detail. In fact, we have three estimators of *βi*: the estimator from the unrestricted model, *β*<sup>ˆ</sup>*iu*; the estimator from the restricted model (where *βj* = 0), *β*<sup>ˆ</sup>*ir*; and the estimator after a preliminary test,

$$b\_i = w\beta\_{i\nu} + (1 - w)\beta\_{i\nu} \qquad w = \begin{cases} 1 & \text{if } |t\_j| > c\_{\nu} \\ 0 & \text{if } |t\_j| \le c\_{\nu} \end{cases}$$

for some *c* > 0, such as *c* = 1.96 or *c* = 1. The estimator *bi* is called the pretest estimator.

The estimators *β*ˆ*ir* and *β*<sup>ˆ</sup>*iu* are linear and (under standard assumptions) normally distributed, but *bi* is nonlinear, because its distribution depends on a random restriction. The pretest estimator is therefore much more complicated than the other two estimators. But it is the pretest estimator that is commonly used in applied econometrics, because in applied econometrics we typically use *t*- and *F*-statistics as diagnostics to select the most suitable model. That in itself is not ideal, but what is worse is that we typically ignore the model selection aspect when reporting properties of our estimators.

The pretest estimator is kinked and therefore inadmissible. Its poor features are well-studied; see for example (Magnus 1999). Surely we should be able to come up with an estimator which performs better than the pretest estimator. This is where model averaging comes in.

### **5. Model Averaging**

In (Magnus 2017) I tell the following story.

A King has twelve advisors. He wishes to forecast next year's inflation and calls each of the advisors in for his or her opinion. He knows his advisors and obviously has more faith in some than in others. All twelve deliver their forecast, and the King is left with twelve numbers. How to choose from these twelve numbers? The King could argue: which advisor do I trust most, who do I believe is most competent? Then I take his or her advice. The King could also argue: all advisors have something useful to say, although not in the same degree. Some are more clever and better informed than others and their forecast should ge<sup>t</sup> a higher weight. Which way of thinking is better?

Intuitively most people, and I also, prefer the second method (model averaging), where all pieces of advice are taken into account. In standard econometrics, however, it is the first method (pretesting) which dominates.

There are theoretical and practical problems with the pretest estimator. One practical problem is the property that—if 1.96 is our cut-off point—for *tj* = 1.95 we would choose one estimator and for *tj* = 1.97 another, while in fact there is little difference between 1.95 and 1.97. This is not satisfactory.

These and other considerations lead us to reconsider the estimator

$$b\_1 = w\hat{\beta}\_{iu} + (1 - w)\hat{\beta}\_{ir}$$

by allowing *w* to be a smoothly increasing function of |*tj*|. This is model averaging in its simplest form, and we see that it is just the continuous counterpart to pretesting. In model averaging we give weight to all models of interest, but not in the same degree, while in pretesting we select one model after a preliminary test, precisely as the King in the story above.

In practice, econometricians use not one but many models. One of these is the largest and one is the smallest. Neither is probably the most suitable for the question at hand. If we use diagnostic tests to search for the best-fitting model, then we need to take into account not only the uncertainty of the estimates in the selected model, but also the fact that we have used the data to select a model. In other words, model selection and estimation should be seen as a combined effort, not as two separate efforts. This is what model averaging does. It incorporates the uncertainty arising from estimation and model selection jointly. Failure to do so may lead to misleadingly precise estimates.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflicts of interest.
