  Model degrees of freedom ($\df$) is a fundamental concept in statistics
because it quantifies the flexibility of a fitting procedure and is
indispensable in model selection. To investigate the gap between $\df$ and the
number of independent variables in the fitting procedure,
\textcite{tibshiraniDegreesFreedomModel2015} introduced the \emph{search
degrees of freedom} ($\sdf$) concept to account for the search cost during
model selection. However, this definition has two limitations: it does not
consider fitting procedures in augmented spaces and does not use the same
fitting procedure for $\sdf$ and $\df$. We propose a \emph{modified search
degrees of freedom} ($\msdf$) to directly account for the cost of searching in
either original or augmented spaces. We check this definition for various
fitting procedures, including classical linear regressions, spline methods,
adaptive regressions (the best subset and the lasso), regression trees, and
multivariate adaptive regression splines (MARS). In many scenarios when $\sdf$
is applicable, $\msdf$ reduces to $\sdf$. However, for certain procedures like
the lasso, $\msdf$ offers a fresh perspective on search costs. For some complex
procedures like MARS, the $\df$ has been pre-determined during model fitting,
but the $\df$ of the final fitted procedure might differ from the
pre-determined one. To investigate this discrepancy, we introduce the concepts
of \emph{nominal} $\df$ and \emph{actual} $\df$, and define the property of
\emph{self-consistency}, which occurs when there is no gap between these two
$\df$'s. We propose a correction procedure for MARS to align these two $\df$'s,
demonstrating improved fitting performance through extensive simulations and
two real data applications.
