# Is normality of residuals necessary for drawing conclusions from Impulse Response function

I know the issue of normality of residuals has been discussed here quite a lot, and I've learned that there are some cases in which it can be a less important hypothesis to test, while more critical in others (see for example here and especially here - discussion of the ARIMA case in the comments to the OP). Anyhow, non of the posts I read dealt specifically with the meaning of normality of residuals on the impulse response function from ARIMA type models. Thus I'm still confused about the implications of non normality (rejecting Jarque-Berra H0) in this case.

1. ARIMA/VAR models assume residuals to be white noise. meaning: zero mean, finite variance and no serial correlation. this can be checked with a Portmanteau test for example. But do the residuals have to be gaussian also? there is a similar question here, but the answer doesn't refer to the normality part.
2. AFAIK, if W.N is also gaussian, it means that the residuals are i.i.d. I've read somewhere that i.i.d residuals are required for IRF in the case of VAR. As VAR is very much based on ARIMA, I'm inclined towards understanding that this is true for univariate ARIMA too. But since using IRF in ARIMA isn't so common, I couldn't find any definitive answer.

it would be great if anyone here could clarify this theoretically.

Practically, the reason I'm asking is that I'm banging my head trying to fit ARIMA to a bunch of time series, in order to be able find it's underlying dynamics using IRF. I've met all the other assumptions and the fit was reasonable, but I couldn't get the residuals to be normal even with differencing/power transformations and also considering level shifts etc. using tsoutliers::tso() in R. of the data. So I don't know whether I should trust any IRF results.

The data I have is a daily time series (N = 335) of news media coverage of different topics. I assume the reason for non-normality is the fact that this data can have lots of zeros (no coverage of a topic on a given day) and also a small amount of ones (which means a topic was salient on that day) which makes a heavy right tail. treating ones as outliers or replacing zeros with some other value (MA for instance) would miss the whole point.

So what I'd really like to clarify at this point is whether ARIMA and IRF can be used without normality, and if not - what else should I try? In the latter case, I probably should post another question with some data so that "what else should I try" could be answered properly, but any general pointers would be appreciated as well.

• It could be useful to separate the issues of normality (a particular distribution) from i.i.d.'ness (which works for any distribution). If you basically care about normality but not i.i.d.'ness, then normality is not needed. There is nothing in the construction of IRFs that relies on normality. But if you use normality to establish i.i.d.'ness, then the question is essentially about the need for i.i.d.'ness for IRF. Then I would rephrase the question and its title to stress this aspect. And please use capital letters where appropriate. Jun 14 '17 at 7:47
• sorry about the caps. English is a second language to me, so it must have annoyed you much more than it does me :) As for the normality/i.id - I think normality is more of the issue at this point, but I'll consider changing it as I read answers and have a better understanding of the issue. Jun 14 '17 at 15:52
• Now that you have a couple of aswers (and the one by C.H. is pretty good, in my opinion) it might make less sense to change the question so as to move from normality to i.i.d.'ness, because then there would be a discrepancy between the updated question and the answers to the original one. If i.i.d.'ness is still your concern, perhaps that deserves a new post. Jun 14 '17 at 16:02
• my thoughts exactly. Jun 14 '17 at 16:04

As @RichardHardy points out, normality of the errors is not necessary for estimated IRFs. The IRFs are functions of the estimate of the variance-covariance matrix of the residuals and of the VAR slope coefficients, both of which can be shown to follow asymptotic normal distributions under weaker conditions than normality (e.g., finite fourth moments and symmetric error distributions).

Hence, by the delta method, one may establish that the IRF estimates also are asymptotically normal.

That said, if the errors are normal, the asymptotic variance-covariance matrix of the impulse response estimator simplifies. Depending on the software you use, it may be the case that this simplification is assumed, so that it would also be necessary that normality is satisfied for inference (e.g., confidence bands) to be asymptotically valid.

Detailed discussion of these issues is for example provided in Lütkepohl, New Introduction To Multiple Time Series Analysis. See in particular Proposition 3.6 and the subsequent Remark 4.

Normality is required to perform tests of significance and sufficiency. Daily data is better analyzed by using deterministic factors such as day-of-the-week , holiday effects and perhaps even day-of-the-month while incorporating any anomalies/level shifts/local time trends/seasonal pulses and when needed an arima structure.

Forecasting may even require focusing on variability being day-dependent as randomness may not be homogeneous across days.

I am confused by your IRF reflection as correlation and partial correlation statistics are sufficient to aid the characterization OF arima models. To me IRF are similar to regression weights and as such their standard errors can lead to t tests which normally require a parametric test thus normality comes into play with respect to the residuals.

• I'm aware that this may not be the most common usage of IRF - but IRF seems to be suitable for my research question. Suppose I have a time series representing news coverage of some topic during one year. I want to answer the question how does that topic coverage respond to an abrupt change (increase) in it". for example, it can decay quickly, it can go up then down then up again in the course of one week... a similar logic can be found in the first answer here. Jun 14 '17 at 15:47
• I do use day of week dummies and anomaly detection, made available in R 'tsoutliers::tso()'. In some cases it helps me get better AIC than just 'forecast::auto.arima()'. Jun 14 '17 at 15:48
• Do you also detect particular day-of-the-month effects , long-weekend effects , pre and post individual holiday effects . Tsoutliers can be helpful but is a far cry from a complete solution Jun 14 '17 at 16:04
• I admit I didn't check day-of-month yet, nor holidays. I haven't considered it because I'm looking at actual articles rather than user traffic/searches etc., so I guess I though it shouldn't have much effect. I can give it a try. Should this be done with dummies also? Jun 14 '17 at 16:13
• yes .... but you need software to conduct searches in order to ferret out the latent factors/variables ... not easily accomplished ...angels walk where devils fear to tred Jun 14 '17 at 18:27