Can you take the moving average of quarterly data of an explanatory variable in a regression to smoothen noise and get more accurate coefficients? I'm trying to use acceleration of quarterly data on household debt (the difference in the difference in debt) in a regression on unemployment (only concerned with correlation) but quarterly data is much noisier than annual data, which may weaken the correlation. I was wondering if transforming the already differenced (twice) data as a moving average would change my coefficients at all and if they would be more accurate. Is this a common technique in econometrics?
Let's say $D_t$ is the stock of debt at time $t$.

The first difference is $D_t - D_{t-1}$.

The second difference is $(D_t - D_{t-1}) - (D_{t-1}-D_{t-2}) = D_t - 2D_{t-1} + D_{t-2}$.

The [4-period] moving average is then given by

\begin{align*}
&amp;= \frac 1 4 (D_t-2D_{t-1}+D_{t-2}+D_{t-1}-2D_{t-2}+D_{t-3}\\
&amp;\qquad+D_{t-2}-2D_{t-3}+D_{t-4}+D_{t-3}-2D_{t-4}+D_{t-5})\\
&amp;= \frac 1 4 (D_t - D_{t-1} - D_{t-4} + D_{t-5})\\
&amp;= \frac 1 4 ((D_t - D_{t-4}) - (D_{t-1} - D_{t-5}))
\end{align*}

Which would be the yearly increase in the stock of debt this period compared to the same thing a quarter ago. Only you could tell if this is what you want in your regression. Something to consider is only to use the first difference: $D_t - D_{t-4}$.