Regression on time series and its segment series

Question

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)?

Thanks!

Answer 1

I don't think I understand the question. But I'll throw something out there and maybe it'll get others to respond.

if ts_full=0,1,2,3,...;
the breaks happen at T1 and T2

ts_1=0 before T1 and ts_1=ts_full-T1 after T1;
ts_2=0 before T2 and ts_2=ts_full-T2 after T2;
then for y~ts_full+ts_1+ts_2

ts_full = trend prior to T1;
ts_full+ts_1 = trend after T1 and before T2;
ts_full+ts_1+ts_2=trend after T2;
ts_1 = change in trend at T1;
ts_2 = change in trend at T2;

for y~ts_clean;
ts_clean if the trend across all three time periods;
ts_full above is the trend only across time before T1;
if ts_1 and ts_2 = 0 then the ts_clean should be the same as ts_full (the trend in the first time period was representative of all three time periods)