# Forecasting a time series $(x_t,{\bf Y_t})$ where all we care about is forecasting $x_t$

Consider a multivariate time series $$(x_t,{\bf Y}_t)$$ $$1\le t \le n$$ taking values in $$\mathbb{R}^{d+1}$$, and suppose that we wish to forecast $$x_t$$ using its own path as well as the "exogenous" series $${\bf Y}_t$$, for instance using a linear time series model. For the sake of simplicity assume all series are (jointly) stationary. One approach that seems clear is to fit a VAR model, and then use the corresponding forecasts for $$x_t$$, but this seems sub-optimal, since the VAR model fit using ordinary least squares would not optimize the parameters to forecast $$x_t$$ alone.

Transfer function models also do not seem useful here, since, when forecasting $$x_{n+h}$$, they presume the corresponding "covariates" $${\bf Y}_s,\;\;s\le n+h$$ are known when making the forecast, when we would only have observed $${\bf Y}_s,\;\;s\le n$$. I guess one might forecast the $${\bf Y}_t$$ series to input into the transfer function model, but again this doesn't seem optimal. Having done a fair bit of googling on the topic, I could not find any real guidance on what seems like a very standard problem.

Can anyone point me in the right direction? Or perhaps some of these simple ideas (VAR, transfer function model with forecasted covariates) is more optimal than I think?

In response to IrishStat's comment, I'll post a simple example using a transfer function model (at least a simple version of one using simple linear regression with ARIMA errors as implemented in auto.arima). Suppose that we wish to forecast the cardiovascular mortality in Los Angeles county, and we also have access to the daily temperature and particulate matter pollution concentration (data from the astsa package in R). I can do this in R as follows:

#Begin R code#
library(astsa)
library(forecast)
library(TSA)

\#taking data at monthly resolution (every four weeks, so that the approximate
\#seasonality/frequency is 13

cmort2=ts(lap[seq(1,508,by=4),3],frequency = 13)
temp2=ts(lap[seq(1,508,by=4),4],frequency = 13)
part2=ts(lap[seq(1,508,by=4),11],frequency = 13)
dat.mat=cbind(as.numeric(temp2),as.numeric(part2))

\#producing forecasts for temp2 and part2 using auto.arima to be
\#fed into the arimax model

temp.mod=auto.arima(temp2)
part.mod=auto.arima(part2)

temp.for=forecast(temp.mod,h=12)
plot(temp.for)
part.for=forecast(part.mod,h=12)
plot(part.for)

temp.for=ts(temp.for\\\$mean,frequency = 13) part.for=ts(part.for\\\$mean,frequency = 13)
dat.mat.for=cbind(temp.for,part.for)

ar.regf=auto.arima(cmort2, xreg=dat.mat)
x=forecast(ar.regf,xreg=dat.mat.for,h=12)
autoplot(x)

\#for comparison, a simple SARIMA model excluding the covariates

ar.noregf=auto.arima(cmort2)
x.noreg=forecast(ar.noregf,h=12)
autoplot(x.noreg)

##end R code##


1. Is this really the best/a reasonable thing to do to forecast $$x_t$$=LA cardiac mortality? The sub-optimal part seems to be that we must input forecasts for the covariates, which rely on modeling the covariates separately from the response.
2. I assume that the confidence bands produced in forecasting $$x_t$$ are not accurate since they do not account for the uncertainty in the forecast of the covariates. Does anybody know if this is the case? I could imagine fixing this myself by producing confidence bands via simulation, but one wonders how to automatically incorporate the forecast uncertainty into the confidence bands for the transfer function model.
• You say "they presume the corresponding "covariates" are known" . I say "they can be forecasted using their history AND the uncertainty in those forecasts can be propagated to get a composite/collective uncertainty in the output series." Post your representative data and I might be able to help further. One caveat the causal variable is ASSUMED to be not dependent on the output series – IrishStat Mar 22 '20 at 19:38
• Thanks for the comment IrishStat! I added an example, and have asked a more specific question. – LostStatistician18 Mar 22 '20 at 21:52

Does anybody know if this is the case?

Yes your are right it regrettably does not incorporate the uncertainty in the predictors .

You say " I could imagine fixing this myself by producing confidence bands via simulation, but one wonders how to automatically incorporate the forecast uncertainty into the confidence bands for the transfer function model.

I say "Not only did I wonder I actually implemented this feature into AUTOBOX , which I have helped to develop using multi-variable monte carlo techniques allowing for the inclusion of possible future anomalies . I used some innovative convolution procedures to accomplish this tour de force . I refrained from publishing these major advancements due to understandable exclusivity concerns.

I am sure that with enough time and perseverance you will be able to follow my guidance here.

There is an R version available if you are interested.

• I hate to say it, but it deserves being made explicit: "refrained from publishing" along with marketing-speak is a classic snake-oil recipe. Maybe it works, maybe it doesn't, and the only bits of evidence we have are (1) your claims and (2) any public applications of your work. There are some memorable applications of your software that produced measurably inferior results, so what is left? Publish and let your peers validate your "innovative ... tour de force." – whuber Mar 23 '20 at 11:06
• That would be telling .The proof is in the pudding . – IrishStat Mar 23 '20 at 11:56
• That is about as anti-scientific and anti-statistical as one can get. If you want to support your approach through results, then at the least you should publish your forecasts and then, later, analyze their accuracy. The standards applicable to software should be no less stringent than those we apply to fortune tellers and seers. Please note that I am not asserting your procedures are bad or even inferior: I am only saying that they cannot be trusted until such evidence is produced and the scope of their applicability is clearly delimited. – whuber Mar 23 '20 at 13:29
• Hi: There's a paper by Richard Baillie ( econometrician ) whose title escapes me that is related to your question. If it doesn't give the direct answer, it will atleast point to some other good references. I'll try to find a list of his papers and get the title. I can't do that right now but probably tonight. Also, feel free to look for it yourself. I'm pretty sure that I spelled his name correctly. – mlofton Nov 9 '20 at 21:31
• Hi: It was easier to find than I expected. I was referring to the first one right at the very top but, based on the titles, it could be that the second or third one are more relevant. I hope it helps in some way. As I said, one of them should atleast lead to other useful references if nothing else. msu.edu/user/baillie/articles.htm – mlofton Nov 9 '20 at 21:35