So, the model that we've estimated this log-linear regression using the first two years of data has given us a particular equation. Our intercept value 319, our slope value 118 and we're using natural logarithm of the weak number. Well, we can use that to make all of our predictions. What we want to look at next during that calibration period is, are there particular times of year that we're systematically missing. And to do that we're going to look at the residuals, and we're going to take the difference between what did we observe in our data versus what did the model project. What was expected under the model. And if we were to plot out the residuals what do we see? Well, we see those summer months that we pointed to earlier, looks like we're systematically missing in those periods. Perhaps it looks like the end of the winter, looks like we're under-predicting in both of those cases. Seems to be some evidence within our calibration data that there is month-to month-variation and we see that again even in our forecasting period. That those periods seem to stand out where we under-predict in July, we seem to over-predict in let's say, November, December. So this would be a good indication that we need to include month-to-month variation. There's something about July. There's something about December that a model without these time varying factors just isn't able to pick up. So how do we account for that? Well, maybe we say that we need to add in quarters Q1, Q2, Q3, maybe we say that we need to add in month-specific dummy variables. Either of those approaches is going to be aimed at trying to capture that piece that right now we're missing. So we'd done in producing this plot was to say all right, well let's put in month-to-month dummy variables. And when we put in these dummy variables, they're going to enter as x variables. We're going to have 11 dummy variables, and the reason why we have 11 and not 12 is imagine setting all 11 of those dummy variables to zero. Well, what's left over? When all of those are equal to zero, the intercept is actually going to reflect our baseline level. So what these dummy variables are really reflecting when we include them in our analysis is the difference relative to your baseline model. So, supposed= our baseline month is January, I'm going to include a dummy variable for, is it February, yes or no? Well, that February dummy variable, it's reflecting the difference between the baseline for January and the baseline for February. Same thing for the dummy variable for March, difference between January and March. Dummy variable for April, difference between January and April. Doesn't matter which month you set as your baseline but if I've got M different months or T different time period, I'm going to include T-1 dummy variables. Now, the important thing in specifying these time periods, they need to recur within our data. So, we've got two years worth of data, for all of the months we're going to have at least two observations. Right? Because we have weekly data, we actually have more than that. But, when we include these seasonal factors, the assumption is that they're going to recur from year to year. And that's what's going to allow us to use them to predict the future. We're assuming that there's something specific let's say about the month of July. Well I've observed the past two Julys. I'm going to make the assumption that the next July that I haven't observed yet is still going to be different. So If July tends to have higher than average demand for the last two years, I'm going to make the assumption that we're going to have higher than average demand in July this coming year, the year after that as long as I'm trying to make those forecasts. We see when we bring in these monthly dummy variables, we do pretty well. We're now capturing those seasonal patterns better than we had before. Now, you can look out into the future and you can see it's not perfect, but we're coming a lot closer to capturing the patterns that we're observing in the data. Well, what if we look at an alternative? Suppose we had run linear regression saying, let's just use the most recent year of data and include monthly effects there. And those monthly effects only calibrated on that second year of data. Well, our trend line would look, it would probably look something like that. And we're getting the fluctuations around that based on the data. So we can see that just relying on the most recent data, not necessarily going to be better for us. Whereas when we use the last two years of data, found an appropriate specification for the trend model, included the month-to-month variation based on the data. Turns out that it yields much better forecasts for us. Now, using the model that we built, we can make predictions going out into the future. So, we use two years for calibration, can we go beyond our calibration period? Absolutely. We can forecast out as far as we want. And that's what you would keep on doing. Testing the model and seeing when is it time to estimate it. So we had worked with three years of data initially. We said we're going to use the first two years for calibration tested on one year of data. Once we found the specification that was appropriate, let's estimate our log linear regression model on all of the initial data that we have, and now let's try to forecast out a couple of months using that model. And you see from the dotted line here that we're doing pretty well. Not perfect, we still have this slight under-prediction in those summer months. But we've built a pretty good model here in terms of being able to forecast the future, and all we've included is a trend and month specific dummy variables. So there's not necessarily a need to put in so many predictors into your model. Rather, you want to give some thought to what's the most parsimonious model that I can construct. And one that's going to have some staying power that I'm not going to have to re-estimate, necessarily, in the short term.