I was wondering how to accurately get the correlation between a variable of percent change return and a variable of dollar change or basis points. Should I standardize both variables or will that lose relevant information? I would appreciate any guidance.
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2 Answers
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When calculating a correlation it is generally advised to standardize, yes. I don't see why this case would be an exception although failing to standardize wouldn't probably be the biggest issue here. This is because the variance of the two variables are fairly comparable.
It is very important to standardize when you're looking to get a correlation between some percent change and the level yearly revenue of a company in dollars, for example.
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When calculationg the Pearson correlation coefficient, it does not matter. It is defined as
$$ \rho\equiv \frac{cov(x,y)}{\sigma_x\sigma_y}=\frac{\mathrm{E}\left((x-\mu_x)(y-\mu_y)\right)}{\sigma_x\sigma_y}=\mathrm{E}\left(\frac{x-\mu_x}{\sigma_x}\frac{y-\mu_y}{\sigma_y}\right)=\mathrm{E}\left(z_xz_y\right) $$
where $z_x,z_y$ are standardised.
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2$\begingroup$ Agree that standardization with a constant doesn't matter but expressing one variable in dollar vs per cent returns can. $\endgroup$– fesJun 29 at 10:05
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$\begingroup$ E.g. assume you i) calculate correlation when one of the variables is expressed in dollar changes $P_{t}-P_{t-1}$ ii) calculate the same correlation but when the variable is expressed in relative (per cent) changes $\frac{P_{t}-P_{t-1}}{P_{t-1}}$. Now i) and ii) can give different results. $\endgroup$– fesJun 29 at 19:09
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$\begingroup$ Your example is of course correct. Maybe OP needs to clarify their question $\endgroup$ Jun 29 at 20:39