Error statistical modeling and inference: Where methodology meets ontology

Abstract

In empirical modeling, an important desiderata for deeming theoretical entities and processes as real is that they can be reproducible in a statistical sense. Current day crises regarding replicability in science intertwines with the question of how statistical methods link data to statistical and substantive theories and models. Different answers to this question have important methodological consequences for inference, which are intertwined with a contrast between the ontological commitments of the two types of models. The key to untangling them is the realization that behind every substantive model there is a statistical model that pertains exclusively to the probabilistic assumptions imposed on the data. It is not that the methodology determines whether to be a realist about entities and processes in a substantive field. It is rather that the substantive and statistical models refer to different entities and processes, and therefore call for different criteria of adequacy.

Appendix: M-S testing and auxiliary regressions

In light of the fact that the Linear Regression model (Table 1) is specified in terms of the conditional mean and variance:

$$\begin{aligned} \begin{array}{cc} E\left( Y_{t}\mathbf {~|~}X_{t}\mathbf {=}x_{t}\right) \mathbf {=}\beta _{0}\mathbf {+}\beta _{1}x_{t},&Var\left( Y_{t}\mathbf {~|~}X_{t} \mathbf {=}x_{t}\right) \mathbf {=}\sigma ^{2},\ \ t{\in }{\mathbb {N}}, \end{array} \end{aligned}$$

(14)

one can test for any departures from the linear regression assumptions: [1] Normality, [2] linearity, [3] homoskedasticity, [4] independence, and [5] t-invariance, by expanding the orthogonal decompositions stemming from (14) (Spanos 1999):

$$\begin{aligned} u_{t} = \overset{0}{\overbrace{E\left( u_{t}\mathbf {~|~} X_{t}\mathbf {=}x_{t}\right) }}\mathbf {+}v_{1t},\ u_{t}^{2} = \overset{\sigma ^{2}}{\overbrace{E\left( u_{t}^{2}\mathbf {~|~}X_{t} \mathbf {=}x_{t}\right) }}\mathbf {+}v_{2t},\ \ t{\in }{\mathbb {N}} , \end{aligned}$$

(15)

to include additional terms representing potential violations from these assumptions. Whereas the adequacy of the model assumes that $E\left( u_{t}\mathbf {~|~}X_{t}\mathbf {=}x_{t}\right) \mathbf {=}0$, the true error might be non-zero when any of the assumptions [2]-[5] are invalid; similarly for $E\left( u_{t}^{2}\mathbf {~|~}X_{t}\mathbf {=}x_{t}\right) \mathbf {=} \sigma ^{2}$. A particular example of such auxiliary regressions whose terms are only indicative of the kind of terms one could use to seek out any remaining systematic information in the residuals, is:

$$\begin{aligned} \widehat{u}_{t}= & {} \overset{[1],[2],[4],[5]}{\overbrace{\gamma _{10}+\gamma _{11}x_{t}}}+\overset{\overline{[5]}}{\overbrace{\gamma _{12}t + \gamma _{13}t^{2}}} + \overset{\overline{[2]} }{\overbrace{\gamma _{14}x_{t}^{2}}}+\overset{\overline{[4]}}{\overbrace{\gamma _{15}x_{t-1}+\gamma _{16}y_{t-1}}}+v_{1t},\nonumber \\&H_{0}:\gamma _{11}=\gamma _{12}=\gamma _{13} =\gamma _{14}=\gamma _{15}\mathbf {=}\gamma _{16}\mathbf {=}0\end{aligned}$$

(16)

$$\begin{aligned} \widehat{u}_{t}^{2}= & {} \overset{[1],[3],[5]}{\overbrace{\gamma _{20}} }+\overset{\overline{[3]}}{\overbrace{\gamma _{21}x_{t}}}+\overset{\overline{[5]}}{\overbrace{\gamma _{22}t + \gamma _{23}t^{2}}}+\overset{\overline{[3]}}{\overbrace{\gamma _{24}x_{t}^{2}}}+\overset{\overline{[4]} }{\overbrace{\gamma _{25}x_{t-1}^{2}+\gamma _{26}y_{t-1}^{2}}}+v_{2t},\nonumber \\&H_{0}:\gamma _{21}=\gamma _{22}=\gamma _{23} =\gamma _{24}=\gamma _{25}=\gamma _{26}\mathbf {=}0 \end{aligned}$$

(17)

In each case the null hypotheses $H_{0}$ assert that the model assumptions hold, taking us back to (15). The terms beyond $\gamma _{10}+\gamma _{11}x_{t}$ in (16) and beyond $\gamma _{20}$ in (17) represent different types of statistical systematic information that the original model might have overlooked. The interesting upshot of this is that the additional terms represent potential violations, which are expressed in generic terms that represent systematic statistical information already in $\mathbf {Z}$ and do not directly refer to any specific substantive factors. Their statistical significance, however, raises questions about how generic terms such as $t$ and $t^{2}$—which represent substantive ignorance—can be replaced by relevant explanatory variables for substantive adequacy purposes; see Spanos (2010c).

One has reduced the problem of probing for model violations to testing the statistical significance of these additional terms, individually or in groups, using simple t-tests and F-tests (Spanos 1999). A rejection of a null hypothesis indicates departures from the underlying model assumption(s).