Currently, I'm trying to fit a linear factor model to a particular time series in finance. For the sake of discussion, let's assume it has the following form:
Y= \mu + \sigma X + \epsilon. Y is the time series, X is the (unobservable) factor(s), \epsilon is white noise. Goal: estimate \sigma and X (case 1: their dimensions are given; case 2: search for the optimal dimension).
However, I'm quite sure that the white noise in the data might have a very large std deviation (compared to the factor's std deviation itself). When the time series contains many data points across a very long time, the "loud" white noise is not an issue. However, especially in financial data, heteroscesdasticity kicks in naturally when the data is very long.
The change in structure is something that occurs naturally in financial data, in contrast to nature science. So, it's unavoidable to limit the data availability (to some extent). When confronted with "short" time series and "loud" white noise, finding the pattern of the factor is very challenging.
My Dear
You can depened on Durbin certieria with test it if your data have Hetro.problem
also you very usefel to use Robust Method to get a optimal estimators of Linear model
Best Regards
Thank you,
As suggested earlier, there are many solutions when confronted with heteroscedasticity, popular one such robust model may help. However, these methods assume the heteroscedasticity is originated from the error terms. Especially in factor models, this can also come from the factors.
We can be quite sure that the financial system before 2008 and after 2008 are different. Mostly due to stricter banking regulation. When the data shows less volatility, this can be because of structure change of /epsilon or of X. We cannot even sure that it is not the other parameters, e.g. /sigma (I didn't assume this to be constant).
Nevertheless, I appreciate the suggestions regarding the heteroscedasticity.
Thank you Gregory Camilli
your suggestion would assume that my assumption that there is an abrupt structure change pre- and post-2008 to be true, and not a continuous incremental change. So basically i need to identify any possible structure change in X. E.g. in post-2008, the banking regulation is "incrementally" stricter and the banks have couple years to implement them, so it is logical to assume that the effect will not be sudden.
At the end, I slowly believe that ignoring the possible heteroscedasticity is not really feasible.
No I understand your suggestion. In the last sentence I said that ignoring heterosk is not feasible. Because the suggested solutions are concentrated about heteroskedasticity instead of the loud noise (which implies that method to handle loud noise most probably does not exist).
To discuss this effectively:
Let assume, we analyze credit defaults. Typically the default rate is influenced both by some systematic factor (typically interpreted to be the economic climate in general) and some idiosyncratic factor (when we are only interested in the portfolio-level, this can be seen as some error).
The systematic factor (if we are thinking about economic climate in general) should show some structural break in 2008. It is possible, that this is affected by general scepticism or stricter banking regulation. What I meant by "the financial system before and after 2008 are different" is that the standard deviation of X is a (non-trivial) function of t (time). After and before 2008, this function probably has a different structure.
At the same time, I expect the same to happen for \epsilon as well. In this particular case, this "factor" would be a lot more volatile than the global system X itself. Because banks are forced to be a lot more conservative by the regulation, so there is some selection bias playing role in the data. The volatility of \epsilon is therefore influenced by the banks' behaviour (which is influenced by the global system).
Thank you for your suggestions. Gregory Camilli
In optimal cases 15 years, some asset classes may only have 7-8 years. Factor analysis can be done without any problem. However, there is no guarantee that the fit is biased due to the loud noise.
I was not trying to analyze any data. The question is intended to be theoretical, rather than practical. In a very large pooled data, white noise is not necessarily an issue (since the idiosyncratic factors are diversifiable). However, banking regulation requires each bank to do their work on risk quantification and there are incentives to not share data with other banks. Assuming that banks do not (and cannot) work together, they have to deal with data with potentially "loud" white noise.
Even though I specify one particular case (banking regulation and risk data), this issue is not particularly unique to this case. So the question was rather general: if banks are forced to work with "noisy" data, is it even feasible to fit some models? I'm especially interested in linear factor models regarding this issue.
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