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I have a whole set of data on [0,T] with an observation variable y(t), and a feature x(t), the two being univariates with no missing data.

For a given period [t, t+h], I am applying a dynamic linear regression:

y(t) = a(t) + b(t) * x(t) + w(t)   
a(t) = a(t-1) + w_a(t)
b(t) = b(t-1) + w_b(t)

Where w, w_a, w_b are the variance of the last term on these 3 lines (following a centered normal distribution).

What matters is that for window [0, h], I normalize my data (x, y) - aka zero mean and 1 std - and then will have a series of estimated parameters a and b after maximum likehood estimation:

time   a     b
   0   0     0
   1   0.41  0.72
   ...
   h-1 0.432 0.55
   h   0.435 0.567

I am interested in the last couple a(h), b(h) giving the intercept and the slope.

Then I take as training set x(1), x(2) ... x(h+1) and y(1), y(2), .., y(h+1), normalize thse two series, and have as well some best estimations for the parameters a and b. I take parameters a(h+1) and b(h+1).

And so on. By rolling the window, I thus have a set of slopes b(h), b(h+1), ..., b(T) and intercept a(h), a(h+1), ..., a(T)

However the normalization differs between each rolling windows. So that I am not sure at all it is a good way to deduce some behaviour about the time series a and b of the "last estimation".

What would be the best method:

  • still perform some analysis on a and b timeseries despite (x,y) normalisation is different on each window?

  • apply the above by normalizing the data on each window [t, t+h] with the means/std of y and x on [0, T]?

  • applying the above without normalizing the data?
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  • $\begingroup$ What do you mean by "normalize"? Are a(t) and b(t) OLS estimates based on $\{ (y_i,x_i), i \le t \}$, or something else? What's the purpose of this whole exercise? $\endgroup$ – StasK Jul 24 '17 at 18:30
  • $\begingroup$ By normalize I mean that I take each dataset on the window [t, t+h], where t goes over [0,T-h] and I retrieve the mean/divide per the std on each window. I edited the question since you're right, it's unclear. $\endgroup$ – Colonel Beauvel Jul 25 '17 at 8:14
  • $\begingroup$ Why do you want/need to normalize? For situations like these, where you need to change the sample, it seems really counterproductive. (May be you can go back and edit again :) ). $\endgroup$ – StasK Jul 25 '17 at 19:05
  • $\begingroup$ I should have mentionned that if I do not normalize, there are some numerical errors in optim in R dlm package. Indeed Y and X have different magnitude (Y around 100000 and X around 300). $\endgroup$ – Colonel Beauvel Jul 26 '17 at 7:46
  • $\begingroup$ Normalize by a fixed constant, then. Divide Y by 100000 and X by 300. $\endgroup$ – StasK Aug 2 '17 at 2:35

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