Definition: Truncated dependent variables are as follows: A dependent variable in a model is truncated if observations cannot be seen
when it takes on vales in some range. That is, both the independent and the
dependent variables are not observed when the dependent variable is in that
range.
A natural example is that if we have data on consumption purchases, if a consumer's willingness-to-pay for a certain product is negative, we will never see evidence of it no matter how low the price goes. Price observations are truncated at zero, along with identifying characteristics of the consumer in this kind of data.
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