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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?

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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*} &= \frac 1 4 (D_t-2D_{t-1}+D_{t-2}+D_{t-1}-2D_{t-2}+D_{t-3}\\ &\qquad+D_{t-2}-2D_{t-3}+D_{t-4}+D_{t-3}-2D_{t-4}+D_{t-5})\\ &= \frac 1 4 (D_t - D_{t-1} - D_{t-4} + D_{t-5})\\ &= \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: $D_t - D_{t-4}$.

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The word "noise" implies that there is error in the measurement of household debt, but there is no good reason to believe this is so. In fact, I argue that a finer time period would give you more detail, and a more accurate measurement of the correlation between two variables. You don't want to find a moving average because that throws away valuable variation in the data.

What you should be more concerned about is which lag is most correlated with the current unemployment rate, and you can find those coefficients by including multiple degrees of lag in the multivariate regression.

Overall, no, it is not a common technique in econometrics. Don't do it. Rather use the standard errors to determine how accurate your estimates are.

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