In the previous videos you have learned how to use a regression model. Predictions using a regression model typically concerned the future. But it can also occur that one wants to forecast the past, or better, put to recreate observations that might have been lost or that do not yet exist. Consider for example the next figure which presents the annual inflation data in Suriname, which is a country in South America just north of Brazil. Here the data available for 1961 to 2015. Suppose now that one is interested in the inflation rate say, at the end of the 19th century. How would you want to create those data? One way would be to search in manuscripts of stories, novels and what have you, to see if there is any anecdotal evidence. A simple alternative idea is to associate inflation with a price of a product that existed also before 1961. That would have the same meaning and use across decades or even centuries. One such product might be a postage stamp. In fact, one of the nice feature of stamps is that the price of the product is printed on it. Also, one might believe that the product is used for similar purposes for all these years. Percentage price changes in stamps for Suriname when averaged over all issued stamps in each of the years, also from 1961 to 2015, is given in this graph. The two graphs with inflation data seem not immediately helpful in particular and there are a few exceptionally large observations. In the data set available to this case. You can see that these concern the years 1993 and 1999. The nice thing though, is that we have the inflation rates for the stems back to 1873 as you can see from this figure. When we write annual inflation as y_t and the annual inflation and stamps as x_t, when we allow for the possibility that maybe a one-year delay denoted as x_t minus 1, a potentially relevant model to predict inflation could be this one. This model is so-called multiple regression model as it includes more than one explanatory factor. The multiple regression model extends the simple regression model by assuming that it is more than a single explanatory variable and that in fact now key such variables. For actual data, the so-called multiple regression model then reads like this. But similar assumptions as for the simple regression model, least-squares estimates for the parameters can be obtained. Similar to the simple regression model, we have the following result; with S_j is the standard error of B_j. The t-test each time concerns a single explanatory variable. It is also possible to test the joint relevance of more than a single variable. For example, you may want to compare the full model against a smaller model. Where in the latter model the third and fourth variable are not included. You can also test the relevance of the entire model by testing the four variable model against this one. When the residual sum of squares of the larger model F for full is written as follows, and the residual sum of squares of the smaller model which has g less variables, is written like this, where the R is for restricted, then the joint F-test for the null hypothesis that both models are equally accurate is given by this expression; where F, g and minus k is an F-distribution with and N minus k degrees of freedom. When we apply the least squares technique to the model for inflation with inflation for stamps this year and that inflation in the previous year, where the observations for 1993 and 1999 are dismissed, we get for 53 observations, the following result. The R-squared of this multiple regression model is 0.960. Deleting all variables at the same time gives an F-test value of 606 which is way beyond the five percent critical value because this critical value is 3.179. So the fit between the two inflation series is quite good. Extrapolating, backwards in time, using the rule on this slide results in recovered inflation data in this figure. So you see, that the simple regression model can be extended with additional variables and for this question on past inflation rates we can use the model to make predictions for the past.
