stayed in hospital for 5 days, they are more probable to have a longer stay.

Remarks and Conclusion {#conclusion}
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In this paper, we proposed a maximum full information likelihood procedure to estimate the Cox model with temporal covariates. The most significant advantage of our procedure is that it can generate well-performed estimation without requiring the extra longitudinal observations before the time to event. There are also three potential extensions to the current work.

Censoring
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Although censoring is not discussed in the current framework, it can be added in the standard way such that censoring is (1) independent from the occurrence of the interested event, or (2) conditionally independent from the event given the covariates at the observational time. In both of the two cases, the consistency and asymptotic normality of the resulting estimators still hold and their proof is straightforward from the proof of Theorem \[theorem-consistency\] and \[theorem-ALASSO\].

Forecast Long Term Survival Rate
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In addition to the hazard function , the estimators proposed in section \[likelihood function\] indicates a consistent estimator to the long term survival rate (LTSR): $$\label{LTSR}
S\left(z,t,t'\right):=Prob\left(T\in \left[t,t+t'\right)\mid Z_t=z\right).$$ Using the notation $g^{-1}$ as in Eq. , the estimator to can be given as 