Solved – When is first differences for time series trend removal appropriate to use

autocorrelationcorrelationdata transformationtime seriestrend

I was just wondering when it is appropriate to use particular time series trend removal methods, specifically first differences and link relatives methods. I have two time series as given here:

I then took the cumulative sum of each series, which now yields two new time series as follows:

It is clear that these two time series look very similar, so I now wish to find just how closely related they are – as such, I would like to try and find the correlation between the two of them. Now, according to Avoiding Common Mistakes with Time Series I should try and remove the trend first before finding correlation. The two (nonparametric) methods that the link above suggests are first differences method and link relatives method. Now, since the two plots appear to be autocorrelated, first differences does not appear suitable (at least I do not think it is, I'd like to include the images but I'm restricted to only two unfortunately). Does this mean that I should instead change to link relatives method? Or alternatively, should I instead use the original time series rather than the cumulative sum time series and then try using first differences?

Best Answer

When is first differences for time series trend removal appropriate to use?

If you are dealing with cumulative sums of stationary series, differencing is a natural transformation to perform. Cumulative sums are nonstationary, have infinite variance and thus generally misbehave when used in linear regression, correlation analysis and similar.

One exception where differencing removes valuable information is in the context of cointegrated time series. That is, when a few integrated series share a common stochastic trend (or a few), this commonality will be removed by simply differencing each of the series, and linear models built by ignoring cointegration will suffer from omitted variable bias due to the omitted error correction term.

Meanwhile, taking the first difference of a stationary series (or a stationary series plus a deterministic time trend) is a rather redundant transformation. It introduces an integrated moving average component in the resulting transformed series which increases the variance in linear models as compared with linear models for the original series in their levels (adjusted for a deterministic time trend, if any).

would it be equally appropriate for me to take first differences on the original time series (as given above)?

The paragraph above should answer this.

Now, since the two plots appear to be autocorrelated, first differences does not appear suitable (at least I do not think it is <...>).

Autocorrelation and first differencing are tangential, except under the presence of a unit root when autocorrelation is extremely high and differencing is a natural transformation. But in general, you may have something like an ARIMA(p,1,q) model which would require differencing but still exhibit autocorrelation due to the AR and MA terms.

Edit (responding to a comment)

but what if I don't know if my original time series are stationary?

You can test for various forms of nonstationarity. E.g. you can test for a unit root (which is one form of nonstationarity) by the augmented Dickey-Fuller test. You can also test for structural change etc. Also, sometimes you will notice visually that the series behaves differently in different periods, which is an informal indication of nonstationarity.

is there an easy way to check if two time series are cointegrated? The tests I've seen so far seem to require checking for statistical significance etc. so I was hoping for a slightly easier and more efficient way.

Perhaps the best you can do without formal testing is running a regression of one series on the other one (or other ones) and visually inspecting the residual. If the residual looks stationary, the series are cointegrated. But also formal testing is not that difficult now that there are functions in statistical packages that do that.

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If you use lm then you should check the residuals to see if they are autocorrelated or not. I guess they are not uncorrelated and hence your t-test are not valid (this is true also for the case of summary(lm(y~t+I(t^2)). This is basiacally beacuse there is a time variable involved in your lm.

I recommend to use Generalized Least Square approach in order to test the quadratic effect and take into account the autocorrelated problem. For example if you assume the autoregressive of order two (see below) for the residuals of your lm (i.e. $e_t=\phi_1 e_{t-1}+\phi_2 e_{t-2}+\nu_t$, where $\nu_t$ is white noise), then the code would be like

library(nlme)
m1=gls(y~t+I(t^2),correlation=corARMA(p=2))
summary(m1)

Note: You should model the error terms correctly first (i.e. finding the order of $p$ and $q$) maybe by ckecking the ACF or PACF of the residuals in your lm. In above, I assumed AR(2). More complicated ARMA model can be considered and tested.

Solved – First remove seasonal trend or long-term trend in time series

As you suggested you one can observe non-stationarity (symptom) but the correct remedy (medicine) is unclear. The correct remedy could be multiple level shifts , multiple trends , seasonal pulses too name a few. Assuming any one approach is both simple and potentially damaging to good statistical analysis.. The high road is to to "listen to the data" ala Bacon,Box,Tukey et al and form the appropriate form of non-stationarity adjustment (much like a good drug prescription) to render the data stationary without incurring damage. The whole idea is to keep the model simple but not too simple.

Non-stationarity can be induced by changes in model form , changes in parameters, changes in error variance besides what has been previously listed here. The message is avoid presumptive cook-book rules suggested by some textbooks and commentators particularly x-11 and it's variants and use the best statistical tools available to form/identify usable solutions.

For example review the outliers and changing error variance when using x11 on the classic airline series to construct the irregular or error process.

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