# Seasonal differencing a when the sample size is small when estimating a VAR

When estimating a VAR the series must be level and seasonally stationary. However I have only 48 data points. I first made the series level stationary based on the ADF test and performed HEGY test for seasonal unit roots. Few series which were level stationary exhibited seasonal unit roots. I am unable to take the seasonal difference due to the small sample size. How to deal with this issue? I have included a plot of transformed series (by season) submitting to HEGY test below:

• Consider including a plot of the series (specifically, the transformation that you are submitting to the HEGY test). Perhaps an "eye test" will be more telling. Jan 21 at 18:26
• @RichardHardy i edited the question by including plots Jan 21 at 18:32
• Could you perhaps draw each season in different color, so that there are four different-colored lines in each graph? If that is not too difficult to do. Jan 21 at 18:34
• @RichardHardy I edited the Q by including each season by a different color.(monthly) Jan 21 at 18:38
• Hm, I am sorry, I cannot see much. Perhaps a picture of the type I have included would do it? Not sure... I have edited my answer to include some more thoughts. Jan 21 at 18:44

Are you sure the series have seasonal unit roots? Given seasonal integration, neighboring observations (that obviously belong to different seasons) may diverge from each other as the sample size grows. Here is a simulated example of a seasonally-integrated quarterly series: Is this what you are observing?

The null hypothesis of HEGY is presence of a seasonal unit root. Given a small sample, the power of the test is likely low, so it might just be struggling to reject a false $$H_0$$. Since your sample is only 4 observations per season, I would not trust the result of the test. While the test statistic is likely derived using an asymptotic approximation, 4 is quite far away from $$\infty$$... And this is not specific to HEGY alone; every test needs data to be powerful. (Well, there is an exception to every rule, but you get the general idea.)

I am not saying your series cannot have seasonal unit roots, but there is likely too little data to conclude anything much about it.

Regarding

When estimating a VAR the series must be level and seasonally stationary

see Ashley & Verbrugge "To difference or not to difference: a Monte Carlo investigation of inference in vector autoregression models" (2009). Here are some notes of the paper I took for myself:

1. For model specification (short-term Granger causality) testing, use either a VAR in differences model or a lag-augmented VAR model (a levels VAR with lagged differences in addition to lagged levels); either has reasonably accurate size and comparable power in our simulations.
2. Estimate the IRFs (and confidence intervals for them) using a model in which the dependent variable is in levels – using either the levels model or the lag-augmented VAR model – with a trend term included. The bias-corrected bootstrap confidence intervals appear to be somewhat preferable to using asymptotic standard errors. The actual coverage of nominally 95% confidence intervals may be substantially less than 95% except for very large samples, especially past lags one or two.
• Yes, according to HEGY test the some series contain seasonal unit roots. Is there a test other than HEGY test to be used in small samples and papers which describe the low power of HEGY test in small samples? Jan 21 at 18:28
• Thank you for the clear explanation! I will check the paper and try to get some more insights. Jan 21 at 18:48
• @Geek_Tech, while I surely appreciate my answers getting accepted, you do not have to rush. Perhaps someone else comes along and provides a better answer. They might be more tempted to do that if the existing answer has not yet been accepted. Jan 21 at 18:49
• thanks for the suggestion. Il keep looking for some other suggestions and accept afterwards :) Jan 21 at 19:06