[SOUND] The question that we address in this video concerns safety. Of course, we would like our surroundings to become safer, and many policies are geared towards increasing safety. Consider, for example, the data on casualties at railroad crossings in the Netherlands for the years 1975 to 2015 in this picture where a downward trend is obvious. Of course one would want to have this number as small as possible. For this perhaps not likely that the zero casualties will ever occur nor is a negative number of casualties possible. So when we will look at the trend in these data, we need to account for that later on. The simple regression model that can be fitted to the data with a trend is this model. Where T runs from 1 to capital T, where T is what is called a deterministic trend. Ignoring the epselon T, then the first, say, four observations are these. When delta is smaller than 0, then with each time unit, the variable, gets smaller with that value delta. Now, how about this model? Again, ignoring the epsilon T and taking Y oneness to first observation, then the first four observations here are those. So, also here, each time the variable gets smaller with the value delta when delta is smaller than 0. Then meaning that also this model can be considered when predicting a variable with a trend. Given that the two models each describe data with a trend, that is common practice to make a choice between the two by looking at the multiple regression. And to see if beta is equal to 1. If that is the case, then usually there is no need to include the deterministic trend because there is already 1 there. To use a typical t-test for this purpose where usually the focus is on whether a parameter is 0, we rewrite a multiple regression as follows. Where rho now is equal to beta minus 1. The t-test on rho as 0 is not distributed as standard normal. At the same time, we're usually not interested in the two-sided case with only in the one-sided case that beta is smaller than 1. The 5% critical value appears to be around minus 3.4. And hence when the t-test value is to the left of minus 3.4, we say that we reject the null hypothesis that beta is 1. And hence we will then resort to a model like this one. If you would have found that beta as 1 cannot be rejected using a statistical test, we would have continued with this model. Now, let us see how this works out for the casualties, where the data are transformed using the natural log transformation. Least squares for 40 observations gives for the parameters alpha, rho and gamma in this model the following results. Using r, the t-test on rho is equal to 0 equals minus 4.572. And this is more negative than a critical value of minus 3.4. So we can conclude that for the casualties data a plausible model reads as follows. Where least squares estimates are for a and c as before, but now we have a different result for rho because beta is rho plus 1. As c is minus 0.038 with standard error 0.009, there is indeed a significant downward trend. Because 0 is not in the 95% confidence interval. Now, where do the data go to if time and the trend proceeds? Suppose we are at time t, which here is 2015 and the value in the model is then 41. And suppose you want to predict the observaton at t plus 1, this works as follows. The actual observation at t plus 1 is this one where we shifted the subscript with 1. At time t, you do not know, epsilon t plus 1, and it is set equal to 0. So, the forecast is done, simply this. This is for one step ahead. But something similar can be done for two steps. Given the model, the actual observation at t plus 2 is equal to this. And the prediction would then be the following expression. But as you do not know why t plus 1 at time T you plug in its forecast like this. And for T plus 3, T plus 4 and further you can make similar computations. If we use this for the casualties data, I'm back transforming the forecasted observations by taking exponential values. We see that the forecast seem to go somewhere around 8 in the next years, look at the picture. Which says that more safety is to be expected indeed. This concludes this lecture, join me to the next videos with some more questions and methods. It's getting better and better.
