Hi,

People told me to post here, but I am not sure if this is the correct place to report this. Let me know if I should direct it somewhere else.

**TL;DR:**

I think line 157 of <matlab/kalman/likelihood/univariate_kalman_filter_d.m> should be updated to guard against the lost of positive definiteness of the variance matrix when you run a regular (non-square root) kalman filter. For a short-term fix, you can just check whether the quantity is negative before checking whether it is zero, and give a large penalty to the likelihood value when it is negative.

**Detail:**

My colleagues have been having difficulties with estimation whenever they tried to include unit roots. They always found that it would find one point where the likelihood value is good, but the likelihood value falls off a cliff if you try to deviate any small amount from the parameters found (and the model behaviors were terrible for the parameters values found).

I believe I have isolated it to a potential numerical problem interacting with the implementation decision on line 157 (the line where you check whether the observation is informative in the univariate Kalman filter) of the file I mentioned above. The problem is when you update the variance matrices directly (instead of update the sqaure-roots), you can (numerically) lose the positive-definiteness of the variance matrix. This is why there are square-root Kalman filters. This numerical problem interacts with line 157 as it checks whether you have rank zero by check the raw value (not absolute value) against a small number. Hence, when you lose positive-definiteness through numerical problems, it will update the likelihood as if it is equivalent to no observation, i.e., it is going to give it too high value for the likelihood. Hence, the optimization/estimation will active find numerically ill-behaved regions if there are any.

I do not have an exact replication problem, but I have tested a numerically guarded implementation on problems that my colleagues told me they had problems with in the past. I get a nice posterior distribution even around unit and explosive roots. It is not an exact replication because I didnāt run things on Dynare side to confirm, but colleagues have been running into the same behavior at my institution for a decade or so: Whenever they tried (Canova, 2014) with unit-roots in the unmodelled block, the estimation didnāt āworkā with the behavior I described above. My test was on one of the models where they have tried and failed to introduce unit roots.

In the long-term, I believe a diffuse univariate Kalman filter with square-root updating should be implemented. I have implemented (Bierman-Thornton, 1977) for the square-root updating, and thatās the implementation I have tested with. I can probably contribute my code if needed as well (after cleaning things up. But it is a clean(?) implementation of BT1977 anyway, so it can be implemented directly from that paper). In the short-term, you can address this buy checking whether you lost the positive-definiteness and quit the Kalman filter early and return a bad likelihood value (with a disclaimer that this short-term fix is coming from theory and I havenāt tested it. However, the fix will only affect the estimation *only if* you lose the positive-definiteness due to numerical problems).

PS- Disclaimer: I traced the if-else tree to confirm that the diffuse kalman filter case would end up in that file in Dynare 5.6, but I havenāt traced the if-else tree in Dynare 6.0, so things might have been indirectly fixed in version 6.0. I have confirmed that that line remains in Dynare 6.0.