# Is it safe to assume inflation rate and treasury yields are stationary?

I have YoY percent change in CPI and the nominal 10 year Treasury yield.

I want to run some correlation analysis between them but worry they are not stationary. I ran a DF test and found that, assuming no drift or trend, both series are non stationary (but close to stationary).

Is it a huge problem is I start running correlations on the the original values (inflation rates, and yields)? Should I use the first differences in my correlation models instead?

The results are more interesting when I use the levels instead of the first differences, so I’d prefer to use the levels.

Is it a huge problem if I start running correlations on the original values?

Both the raw/untransformed CPI and treasury yields are widely known non-stationary time-series processes:

• The article of Hall et al. (1992) (p. 117 top-right) provides in a footnote, a list of articles that describe treasury yields as an $$I(1)$$ process (making it an $$I(0)$$ process by first-differencing) in an ARIMA setup.

• There are numerous non-famous articles and websites that describes the raw CPI index as a non-stationary process, see here, here and here. The latter article argues that the CPI index is a stationary process under first-differencing. If you find that the YoY percent change CPI is still non-stationary, try to difference the log-CPI of the raw time-series and see whether it yields a stationary process.

In general, when dealing with non-stationary time-series, it will be wise to do your correlation analysis on the stationary processes instead of the levels, so you don't end up with spurious correlations. Also, the sample correlation on stationary processes converges in probability to the true correlation coefficient, $$\hat{\rho} \overset{\mathbb{P}}{\rightarrow} \rho$$ when $$T \rightarrow \infty$$. In general, any sample moment on stationary processes converges in probability to a constant, thus making them interpretable and understandable (Good sources for more information about this: Post 1, Post 2).

I hope my answer provides a bit of insight.

• Thanks, Pleb. Do you by chance know about the following? I read that if you stationarize dependent variable data and obtain a beta estimate using some independent variable… that beta estimate (regression coefficient) can be interpreted as if we did the regression on the levels. So instead of a beta = 2 meaning “a one unit change in the change of x implies a 2 unit change in the change of y” we can simply say “a one unit change in x implies a 2 unit change in y.” Is that true? Dec 9, 2021 at 2:03
• Do you have the source for this? I cannot tell you whether this is true or not (or if it's true under certain scenarios/assumptions). Often in macroeconomics, regressing growth-rates (first-differencing and dividing with lagged value) have just as much economic interpretation, as the interpretation on the original time-series. Thus, if you don't find a solid answer to the above, I would refrain from making any interpretation of the $\beta$-coefficient on the levels and simply apply the intuition on the stationary time-series instead (eg. the growth-rate).
– Pleb
Dec 9, 2021 at 11:33
• @Pleb, I have different opinion on the presence of unit root in these time series. I invite you to take a look. Dec 9, 2021 at 12:26
• Thanks Pleb. Do you have a sense of the easiest way to interpret differenced coefficients? I know I can say “a one unit change in the change of x implies a two unit change for a change in y.” … or I could use the word “growth” but that presents a problem when considering scenarios when x declines and then declines again but at a lower rate (deceleration)…. In that case there is no “growth” so I’m unsure about the best way to with words explain the coefficients. Dec 9, 2021 at 16:50
• I will try onto find the sources that said the same beta is for original and differenced data. Dec 9, 2021 at 16:50

Since you have mentioned taking first differences as a possible remedy for nonstationarity and since this remedy is mainly appropriate when dealing with I(1) processes, let me focus on whether these time series are I(1) or perhaps I(0).
(I(1) implies nonstationarity while I(0) does not imply it but permits it.)

If YoY % change in CPI and the nominal Treasury yield were I(1), i.e. contained unit roots, they could wander off to $$+/-\infty$$ and never return. This is clearly not the case as e.g. you have strong economic arguments against a scenario where the nominal Treasury yield is negative and large, and you would not think inflation or deflation can grow without bound and never come back. Thus, these processes do not contain unit roots.

What about modelling these processes as if they contained unit roots? If you work with relatively high frequency data on the processes, you may find rather strong persistence, suggesting that you can approximate the processes reasonably well using unit-root models. They would work fine over not-too-long time horizons. If you work with low frequency data, the persistence is less strong, and unit-root models do not approximate their behavior that well. A stationary model may then be more useful. Hence, what model you choose could depend on the frequency and time span of the time series.

• Interesting but now I’m entirely confused. When I run all three DF tests (no drift, drift, drift and trend, augmented), I find there is a unit unit. So I’m unsure now. Maybe I will craft a specific question for this.l to get more input. Dec 9, 2021 at 16:46
• @Jason008, your tests do not have enough power to reject a false $H_0$ of presence of a unit root. Dec 9, 2021 at 17:14
• @Jason008, that depends on the frequency and the time span. Could you update your post to include a graph of your two time series (raw)? Dec 9, 2021 at 17:43
• Richard, I’ve updated. I’m unsure how anyone could say these series are stationary. Their means and variances clearly change over time. Please explain why you think they are not stationary. I’ve been reading and see there actually tons of debate about this… one side thinks stationary, the other not. Dec 9, 2021 at 20:22
• @Jason008, if you need this for a school project, it may be safest to do what everyone else does. If they take first differences, you could do the same. You could also tentatively allow the residuals to follow an MA(1) process; I expect a negative coefficient there; if it is small, ignoring the MA(1) structure in the actual model will not hurt much. Dec 10, 2021 at 6:44