# Exploring relation between time series

I have the below time series data:

I want to explore the relationship between dependent variable y and the independent variables x1 and x2. My aim is not forecasting. Just finding the relationship between these variables and making simulations by changing the dependent variables. I know that, y have some other components too, other than x1 and x2. But it is impossible to find these variables for me. So, I have to do my best using these two dependent variables. The data is monthly data. And it can consist of seasonality.

How can I do such an analysis? I don't want an arima model. I mean, I don't want a model which includes dependent variable's lags and error term lags. Which method should I use? Is it enough just to transform these variables into stationarity variables and trying different models to find the best one?

I will be very glad for any help. Thanks a lot.

Data:

period y x1 x2

201401 184 2.23 3.03
201402 194 2.22 3.03
201403 200 2.22 3.07
201404 201 2.13 2.94
201405 184 2.10 2.88
201406 204 2.13 2.89
201407 199 2.13 2.89
201408 200 2.17 2.89
201409 197 2.21 2.86
201410 194 2.27 2.87
201411 196 2.24 2.80
201412 191 2.29 2.82
201501 208 2.34 2.73
201502 203 2.46 2.79
201503 212 2.59 2.81
201504 199 2.66 2.86
201505 200 2.66 2.97
201506 193 2.70 3.03
201507 200 2.69 2.97
201508 206 2.86 3.18
201509 219 3.01 3.38
201510 222 2.94 3.31
201511 233 2.88 3.10
201512 242 2.93 3.18
201601 262 3.02 3.28
201602 260 2.94 3.26
201603 254 2.90 3.22
201604 238 2.83 3.21
201605 241 2.94 3.32
201606 242 2.93 3.29
201607 263 2.97 3.28
201608 238 2.96 3.32
201609 247 2.96 3.32
201610 250 3.08 3.40
201611 267 3.27 3.53
201612 262 3.50 3.69
201701 302 3.73 3.97
201702 286 3.68 3.92
201703 275 3.68 3.93
201704 284 3.66 3.92
201705 294 3.57 3.94
201706 288 3.52 3.95
201707 288 3.57 4.10
201708 301 3.52 4.15
201709 300 3.47 4.14
201710 306 3.67 4.32
201711 307 3.89 4.56
201712 322 3.85 4.55
201801 324 3.78 4.60
201802 331 3.78 4.67
201803 310 3.88 4.79
201804 317 4.06 4.99
201805 330 4.41 5.22
201806 360 4.64 5.42
201807 354 4.76 5.56

I took your series ( length =54 ) and used Transfer Function identification to automatically sort out the impact of the two predictor series and to identify possible omitted deterministic structure. There are a few anomalies (5) that were identified. here is the equation .

The Actual vs Cleansed highlights the identified anomalies . The model statistics are here

THe residuals from the model are free of structure suggesting sufficiency.The ACF of the residuals is here

The Actual/Fit and Forecast graph is here with Actual and Forecast here .of the forecasts for the next 12 periods is here

There is a need for just 1 seasonal pulse a January effect not 11 as had been suggested.

Also good analytics detect any needed differencing ( as in this case ) and correct lag structures while remedying anomalous data points (which now should be investigated !)

see how Excel compares to this ...

For good reading and learning ...https://stats.stackexchange.com/search?q=transfer+function+models

• Thanks a lot @IrishStat for the detailed explanation. Do you know any package in R, which I will able to the similar analysis, where can the anomolies and seasonlity will be detected automatically? Thanks a lot again for sharing your time and for the very useful explanation. – oercim Aug 15 '18 at 3:20
• And when, I look at to the actuals and fits graphs, I suspect about one lag. Maybe,would it be better to include one lag month of the dependent variables? Thanks a lot again. – oercim Aug 15 '18 at 3:36
• AUTOBOX which I have helped to develop has an R version with all the required optional automation . Note that in this case the "seasonality" that was detected was simply a seasonal pulse for the month of January . The appropriate lag structures in this case were Y lag1, X1 contemporaneous and X1 lag1 , X2 lag1 and X2 lag2 – IrishStat Aug 15 '18 at 7:56
• I guess AUTOBOX is not a free tool. Is there any other Package that you know that is free? – oercim Aug 16 '18 at 4:12
• None ...that I know of that treats both causal models and Intervention detection – IrishStat Aug 16 '18 at 5:45

Linear regression, where you take period and turn it into 11 monthly dummies (leave one out as a reference) and so you end up with 13 X's. Super simple, nothing you said makes me think you need a time-series method or something more complex than linear regression at this point. And you can do all of this in Excel (or any other statistical software).

• Don't I have to transform them into stationary variable? – oercim Aug 14 '18 at 20:56
• no. if there's a trend in Y then you may want to add a 14th X with a simple count (can just start at 1). – Chris Umphlett Aug 14 '18 at 21:01
• Thanks a lot for sharing your time to my problem. But one last question. I am confused with that: With this way, won't it be a spurious regression? – oercim Aug 14 '18 at 21:06
• And also maybe, the dependent variable give reaction to the ndependent variables with some lags? So don't i have to need to add lag of independent variables to the model? – oercim Aug 14 '18 at 21:12
• a spurious regression is when two things that are truly not related are found to have a statistical association or correlation. eg, ice cream consumption and crime. Eating ice cream doesn't make people commit crime, but both of them are higher in the summer. – Chris Umphlett Aug 15 '18 at 1:26

I may be blind to something but this seems a basic linear regression model.

If not linear, try taking logarithms and see if that is linear.

And, if you believe there is serial correlation (which looks highly likely), do first differences or first ratios. In other words, don't regress on y(x), but on y(x) - y(x-1) or y(x)/y(x-1). In any normal regime, that should give you stationarity.