# Which test for lagged effect of one time series on another?

I have a data set with three variables: year (21 consecutive years) and two time series which are count data (count1 and count2). I want to know whether count2 correlates with some time delay lag with count1.

Both time series follow a positive (and similar in size) linear trend as resulting by running a linear model for each time series (in R, function lm). According to the Augmented Dickey-Fuller Test (in R function adf.test from tseries) Count1 neither count1 and count2 are stationary but they become stationary after differencing once. According to partial autocorrelation functions of the counts after differencing (in R function pacf) count1 shows significant autocorrelation of order 1 (lag=1) and count2 shows no significant autocorrelation.

Questions:

Should I use a cross-correlation test (in R function ccf) on the variables obtained after differencing each time series (say, diff.count1 vs. diff.count2)?

Or should I use a distributed lag model on the time series after differencing (in R dlm from dLagM)? I have tried but I have problems to select the model with the right time lag because as I increase the value of q (no. of time lags) the AIC improve always (decreases), even when the model is not able to estimate any slope. I know of a third possibility that is the autoregressive distributed lag model (in R ardl from dLagM) on the raw data but I have the same issue with the AIC.

My data look like this (in R):

data <- data.frame(year=1:21,count1=c(7, 40, 86, 4, 73, 199, 400, 673, 1125, 0, 832, 3643, 2236, 2172, 5267, 7228, 0, 6909, 939, 7851, 1231), count2= c(5, 6, 0, 0, 1, 1, 15, 1, 0, 1, 2, 29, 5, 38, 22, 46, 132, 161, 103, 32, 70))

• Time-delayed is also known as lagged, that term can be helpful. – Richard Hardy Oct 23 '18 at 17:06
• you could also investigate Granger causality. – Quetzalcoatl Oct 31 at 17:08

In my opinion one should generally follow https://onlinecourses.science.psu.edu/stat510/node/75/ and Transfer function in forecasting models - interpretation and an example here How to include control variables in an Intervention analysis with ARIMA? using method suggested here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html .

These procedures generally work when there are no outliers/pulses or level shifts induced by an unspecified effect i.e. intervention detection yields negative results and no discernable/significant change in the model's error variance over time see http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html. Analysis detected that both of these Gaussian assumptions needed to be addressed and remedied with this data set.

A simple plot of Y and X visually suggest possible changing activity midway through time

The final model suggests that there is a contemporary and lag1 effect of Count2 the user suggested causal series. .The acf of the residuals suggest sufficiency . The Transfer Function model expressed as a regression model (explicit) is here

A significant change in the variance of the models errors was detected at period 14 suggesting that Weighted Least Squares was needed and here

You asked "Should I use a cross-correlation test (in R function ccf) on the variables obtained after differencing each time series (say, diff.count1 vs. diff.count2)?" . The answer is no as the correct pre-whitening filter is not differencing but a simple ar(1) filter and the resultant cross-correlation analysis is here . Note that due to the untreated level shift at period 12 and one-time pulse at period 13( i.e. bloated error variance ) there is a downwards bias to the cross-correlation test. Determining parameters (p, d, q) for ARIMA modeling discusses this effect on the acf but it equally applies to the ccf. Upon reflection the diiferencing filter you used is essentially an ar(1) with value 1.0 while my filter is an ar(1) with value .754 ... no big issue here !

The "temporary" conclusion that COUNT2 is not important is seconded by this model

As usual I am always interested in other approaches to this data set using "the current methods in vogue" . Finally having both a level shift ( change in the expected value and a change in model error variance at roughly the same point in time ( 12 or 13 ) suggests that this may be a case of transience in the model parameters.

This is a case of Exploratory Data Analysis for time series data where latent factors are waiting-to-be-discovered.

EDITED AFTER CAUSALITY DIRECTION WAS CHANGED ... NOW COUNT2 IS A FUNCTION OF COUNT1

Here is the model ... and the equation .

There is a level shift at period 16 and a pulse at period 18 .

the lags of count1 that are useful to predict count2 are lags 2 and lags 5

The Actual and Fit graph is here

• Thank you @IrishStat for your exhaustive answer. I have just reproduced, in R, the procedure in onlinecourses.science.psu.edu/stat510/node/75 although there is no validation of the assumptions (i.e. constant variance or outliers/pulse - does it refer to outlier in residuals of the ARIMA?). I get a completely different estimate of the autoregressive factor (-0.8466 vs. 0.754). Then, I have tested the cross-correlation of the residuals of the ARIMA on count1 and the filtered values of count 2. No lag results significant, however I have still the issue of violating assumptions. – user3844454 Oct 25 '18 at 16:34
• Following the previous comment. Unfortunately, I have no idea on how to handle the violating assumptions of the ARIMA in R to go on with the filtering and no idea on why those pulses are there. That said, I have one question: when you say "The final model suggests that there is a contemporary and lag1 effect of Count2 the user suggested causal series". My hypothesis is that count1 causes count2 variation so I would expect count1(t) determines count2(t+lag). Did you find count1 and count2 are correlated at lag 0 and 1, the contrary? Where is it in the analyses you have shown? – user3844454 Oct 25 '18 at 16:50
• You said ":I want to know whether count2 correlates with some time delay lag with count1. ... I understood this to mean the count1 was possibly predictable by count2. I will reprocess accordingly.. – IrishStat Oct 25 '18 at 19:12
• and solve for how count1 predicts count2 – IrishStat Oct 25 '18 at 19:39
• Thank you! answers to two questions would help a lot: 1. these time series are count data that I would think to be poisson-distributed but you referred to Gaussian assumptions. I don’t know if this is because the Pearson residuals should be normally-distributed or what else but I don’t feel sure I am getting the point. Does not make any difference these are count data? 2. I have read the link to Pennsylvania Un. and I have learnt quite a few things but I am still not sure to understand which steps you have followed. May you please say them in a schematic way (1,2,3..)? – user3844454 Oct 26 '18 at 15:48