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Working with a panel data set with a daily time series structure I was told to include a lagged dependent variable. The dependent variable is daily electricity consumption of a medium size sample (>200) over a metering period of about 1 year.

In my existing pooled OLS model, outdoor temperature and some consumer specific variables like floor space or number of residents are the most important variables. $R^2$ is around 0.7.

Including yesterday's consumption yields an $R^2$ of about 0.9. However, I am not sure if including the lagged value is meaningful. Today's consumption $y_{t}$ is highly correlated with yesterday's consumption $y_{t-1}$ but this is probably because $y_{t}$ and $y_{t-1}$ are influenced by the same variables (temperature, dwelling size,...). Thus, $y_{t-1}$ and the other independent variables are also correlated which violates some OLS assumptions (if I remember correctly...). Is it useful and in accordance with OLS assumptions to include a lagged value in pooled OLS?

Further, as I have never worked with lagged values before I really have some problems understanding its practical use. There is a similar question in SE but still I don't understand...

When I want to model/predict consumption of day $t$, I will probably not have access to consumption of day $t-1$. So how do I use a model with lagged DVs in practice? Do I just use modelled values so that every $\hat{y}_{t}$ relies on a previously estimated value $\hat{y}_{t-1}$? If so, do I need some kind of starting value for the first day of the time series?

Many thanks in advance for any help in understanding this!

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