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I'm trying to understand how I might best model a variable where over time I've obtained increasingly detailed predictors. For example, consider modeling recovery rates on defaulted loans. Suppose we have a dataset with 20 years of data, and in the first 15 of those years we only know whether the loan was collateralized or not, but nothing about the characteristics of that collateral. For the last five years, however, we can break the collateral into a range of categories which are expected to be a good predictor of the recovery rate.

Given this setup I want to fit a model to the data, determine measures such as the statistical significance of the predictors, and then forecast with the model.

What missing data framework does this fit into? Are there any special considerations related to the fact that the more detailed explanatory variables only become available after a given point in time, as opposed to being scattered throughout the historical sample?

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OK, from experience in using historical data, more history may make the regression fit appear better, but if predicting is the point of exercise, the general answer is be warned. In the case where the data reflects periods for which the 'world' was very different, the stability of correlations is questionable. This occurs especially in economics where markets and regulations are constantly evolving.

This holds for the real estate market also which, in addition, may have a long cycle. The invention of mortgage backed securities, for example, transformed the mortgage market and opened the flood gates for mortgage origination, and also, unfortunately, speculation (there was actually a whole class of no/low document loans called lier loans).

Methods that test for regime changes can be especially valuable in deciding in a non-subjective manner when to exclude history.

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Typically, this can be viewed as a bounded parameter value problem. As I understand your question, you have a less informative parameter (collateral of unknown quality [Cu]) early in your data and more informative (collateral with high [Ch], medium [Cm], or low [Cl] quality) in your later data.

If you believe that the non-observed parameters for the model do not change over time, then the method can be simple where you assume that the point estimates of each are Cl < Cm < Ch and Cl <= Cu <= Ch. The logic is that Cl is the worst and Ch is the best, so when the data are unknown it must be be between or equal to those. If you are willing to be slightly restrictive and assume that not all collateral was either high or low quality during the first 15 years, you can assume that Cl < Cu < Ch which makes it significantly simpler to estimate.

Mathematically, these can be estimated with something like: $$ \begin{array}{lcl} C_l &=& \exp(\beta_1) \\ C_m &=& \exp(\beta_1) + \exp(\beta_2) \\ C_u &=& \exp(\beta_1) + \frac{\exp(\beta_3)}{1+\exp(-\beta_4)} \\ C_h &=& \exp(\beta_1) + \exp(\beta_2) + \exp(\beta_3) \end{array} $$

Where the logit function in Cu restricts the value to be between Cl and Ch without restricting it relative to Cm. (Other functions bounding between 0 and 1 can also be used.)

Another difference in the model should be that the variance should be structured so that the residual variance is dependent on time period because the information within each period is different.

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