An alternative is package resizegather, which shrinks equations of environment gather of package amsmath to \linewidth, if the equation is too large:
\documentclass{article} \usepackage{amsmath} \usepackage{resizegather} \begin{document} \hrule % show text width \begin{gather}\label{model3_coef} y_{t}^{3} = -145.071 - 0.003*x_{t-1}^{7} + 0.459*x_{t}^{6} + 0.001*x_{t-1}^{8} -5.071*x_{t-1}^{9} + 7.322*x_{t-1}^{5} - 0.235*x_{t-1}^{1} \end{gather} \begin{multline}\label{model3_coef_alt} y_{t}^{3} = -145.071 - 0.003*x_{t-1}^{7} + 0.459*x_{t}^{6} + 0.001*x_{t-1}^{8} \\ - 5.071*x_{t-1}^{9} + 7.322*x_{t-1}^{5} - 0.235*x_{t-1}^{1} \end{multline} \hrule % show text width \end{document}

In this case of the page layout, the equation is still much too large for a pleasant outcome and package resizegather warns, if the scaling factor gets below a threshold (default: 95%, can be configured with option warningthreshold):
Package resizegather Warning: Equation line 1 is too large by 92.86534pt (resizegather) in environment `gather' on input line 12.
It should be kept in mind, that the reader should be able to read the equation. Since also the indexes gets scaled down, many readers would need magnifying glasses. Therefore the second equation shows the same equation split in two lines and in its natural size. Package amsmath provides many ways to split an equation.
\resizebox? Also, the parameter needs to be put in math mode:\resizebox{<length>}{<height>}{$ math content $}.\mbox(so takes you out of math mode) but scaling text is always the wrong thing to do