As I'm just starting with permutation tests, I though a question was a good idea. Indeed, thanks to comments by @Glen_b and @user43849, I perceived many misunderstandings and inconsistencies of the theory from my part. For one, I was thinking about testing the magnitude of the coefficient instead of the effect, which is what actual interest.
So, as I'm learning, an actual answer to be criticized sounded just as good.
To answer this question and appoint a permutation strategy that complies with my requirements, I resorted to Anderson MJ, Legendre P. "An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model." Journal of statistical computation and simulation 62.3 (1999): 271-303.
There, the authors do empirical comparisons between four permutational strategies, in addition to normal theory $t$-statistic tests:
- Permutation of Raw Data (Manly, 1991, 1997)
- Permutation of Residuals under Reduced Model (Freedman & Lane, 1983)
- Permutation of Residuals under Reduced Model (Kennedy, 1995)
- Permutation of Residuals under Full Model (ter Braak, 1990, 1992)
Here I'll quote the description given to the strategy put forward by Manly. Given a model $Y=\mu+\beta_{1\cdot2}X+\beta_{2\cdot1}Z+\epsilon$:
- The Variable Y is regressed on X and Z together (using least squares) to obtain an estimate $b_{2\cdot 1}$ of $\beta_{2\cdot 1}$
and a value of the usual $t$-statistic, $t_\text{ref}$ for testing
$\beta_{2\cdot 1}=0$ for the real data. We hereafter refer to this as
the reference value of $t$
- The Y values are permuted randomly to obtain permuted values Y*.
- The Y* values are regressed on X and Z (unpermuted) together to obtain an estimate $b_{2\cdot 1}^*$ of $\beta_{2\cdot 1}$ and a value
of $t^*$ for the permuted data.
- Steps 2-3 are repeated a large number of times, yielding a distribution of values of $t^*$ under permutation.
- The absolute value of the reference value $t_\text{ref}$ is placed in the distribution of absolute values of $t^*$ obtained under
permutation (for a two-tailed $t$-test). The probability is calculated
as the proportion of values in this distribution greater than or
equal, in absolute value, to the absolute value of $t_\text{ref}$ (Hope,
1968)
So this strategy conserves the covariance of the independent variables X and Z. Other methods focus on the testing of partial coefficients in isolation, and these are discussed in the text. Also, possible drawbacks of the strategy of permutation of raw data are given both in the text and in the literature.