I want to test whether segment series explains anything in additional to the full series.
Let's say y and ts_full are time series with same length. And I divide ts_full to 3 non-overlapping sub time series with same length: ts_1 - ts_3. For example, ts_1 has valid values in first time segment and 0 in the rest. Same thing apply to ts_2, ts_3. In this case, ts_full = ts_1 + ts_2 + ts_3
My equation is:
y = b_0 + b_full * ts_full + b_1 * ts_1 + b_2 * ts_2 + b_3 * ts_3 + e
Can I do this? I get very weird result that t-stats for every coefficient is very significant. After second thought, I feel it may break the linear regression assumption that one dependent variable can't be perfect linear combination of others. So I rewrite it as:
y = b_0 + b_full * ts_full + b_1 * ts_1 + b_2 * ts_2 + e
The result then becomes more reasonable. But how should I interpret b_full, b_1 and b_2 then? Can I say b_full is the base coefficient and b_1/b_2 are incremental for sub segment? What's the relationship between b_full and b_clean where b_clean is just simple regression coefficient (y = b_0 + b_clean * ts_full + e)?