**3. Preliminary Test and Shrinkage Estimation**

Researchers have determined that restricted estimation (RE) generally performs better than the full model estimator (FME) and leads to smaller sampling variance than the FME when the UNPI is correct. However, the RE might be a noteworthy competitor of FME even though the restrictions may, in fact, not be valid; we refer to Groß (2003) and Kaçıranlar et al. (2011). It is important that the consequences of incorporating UNPI in the estimation process depends on the usefulness of the information. The preliminary test estimator (PTE) uses UNPI, as well as the sample information. The PTE chooses between the RE and the FME through a pretest. We consider the SUR-PTE of *α* as follows:

$$
\hat{\mathfrak{a}}^{\text{PTE}} = \hat{\mathfrak{a}}^{\text{RR}} I(F\_n < F\_{m,M \cdot T - p}(\mathfrak{a})) + \hat{\mathfrak{a}}^{\text{RR}} I(F\_n \ge F\_{m,M \cdot T - p}(\mathfrak{a})), \tag{10}
$$

where *Fm*,*M*·*T*−*<sup>p</sup>*(*α*) is the upper *α*-level critical value from the central *F*-distribution, *I*(*A*) stands for the indicator function of the set *A*, and *Fn* is the *F* test for testing the null hypothesis of (8), given by:

$$F\_n = \frac{\left(\mathbf{H}\hat{\mathbf{a}}^{\text{OLS}} - \mathbf{r}\right)^{\prime} \left(\mathbf{H}\left(\mathbf{Z}^{\prime}\mathbf{Z}\right)^{-1}\mathbf{H}^{\prime}\right)^{-1} \left(\mathbf{H}\hat{\mathbf{a}}^{\text{OLS}} - \mathbf{r}\right)/m}{\hat{\varepsilon}\_\*^{\prime}\hat{\varepsilon}\_\*/\left(M\cdot T - p\right)},\tag{11}$$

where *m* is the number of restrictions and *p* is the total number of estimated coefficients. Under the null hypothesis (8), *Fn* is *F* distributed with *n* and *M* · *T* − *p* degrees of freedom (Henningsen et al. 2007). The PTE selects strictly between FME and RE and depends strongly on the level of significance. Later, we will define the Stein-type regression estimator (SE) of *α*. This estimator is the smooth version of PTE, given by,

$$
\hat{\mathfrak{a}}^{\text{SE}} = \hat{\mathfrak{a}}^{\text{RR}} - d\left(\hat{\mathfrak{a}}^{\text{RR}} - \hat{\mathfrak{a}}^{\text{RR}}\right) F\_{\mathfrak{n}}^{-1},\tag{12}
$$

.

where *d* = (*m* − 2)(*T* − *p*)/*m*(*<sup>T</sup>* − *p* + 2) is the optimum shrinkage constant. It is possible that the SE may have the opposite sign of the FME due to small values of *Fn*. To alleviate this problem, we consider the positive-rule Stein-type estimator (PSE) defined by:

$$
\hat{\mathfrak{a}}^{\rm PSE} = \hat{\mathfrak{a}}^{\rm RR} + \left(1 - dF\_n^{-1}\right) I(F\_n > d) \left(\hat{\mathfrak{a}}^{\rm RR} - \hat{\mathfrak{a}}^{\rm RR}\right). \tag{13}
$$
