Stationarity, Differencing, and Correlation in Inflation and Treasury Yields

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


# 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.

## Answer by Pleb (score 4)

https://quant.stackexchange.com/a/69026

#### Answering your main question:

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

Short answer: Yes.

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.

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/69030

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.