[SOUND] Welcome at our MOOC lecture series on Econometrics. The topic of this lecture is Simple Regression, and in particular, its motivation. In this introductory lecture, we will consider a simple example concerning the weekly sales of a product with a price that can be set by the store manager. We expect that lower prices lead to higher sales. The econometrician tries to quantify the magnitude of these consumer reactions to such price changes. This helps the store manager to decide to increase or decrease the price if the goal is to maximize the turnover for this product. Turnover is sales times price. In our example, we have two years of weekly data, that is, 104 observations on sales and prices. And the histogram of the sales data look like this. You can see that the majority of weekly sales are somewhere in between 90 and 95 units, with a minimum of 86 and a maximum of 98. Sales of 92 and 93 units are most often observed, each 19 times. The store manager can freely decide each week on the price level, presented on the next slide. This histogram shows that prices vary from 51 to 57, with a median price of 54 and an average price also close to 54. When we plot sales against price that occur in the same week, we get the following scatter diagram. Note, that the number of points in this diagram is less than 104 because some weeks have identical values for price and sales. You can observe that higher prices associate, in general, with lower sales and lower prices with higher sales. You can also see this in this table, where lower prices are associated with higher sales at the left part of the table. If we change the points in the original scatter into the average sales data, we see that the line connecting the average values is roughly linear. And when we go back to the scatter diagram, we see that a straight line can be fitted reasonably well to the points in this diagram. The actual sales are the dots, and the predicted sales are on the line. A given price does not always associate with the same sales level, as there are other factors that also cause variation in sales. For example, the weekly number of customers who do their shopping in this store will influence sales. The observed data are reasonably close to the line, but they do not lie exactly on it. For a given price and a given line, predicted sales is equal to the value of �a� plus �b� times Price. We denote the difference between the actual sales and the predicted sales by the residual e. The coefficient b measures the slope or marginal effect, that is, the change in sales when the price changes with one unit. When the slope is equal to zero, this corresponds to observations, for example, like this. For such data there does not seem to be any relationship between sales and price. Going back to the original scatter diagram, the values of the coefficients a and b are of interest to the store manager as they offer the opportunity to predict sales for a given price. This helps to set a new price if sales are felt as too low. The histogram of sales would just suggest a prediction of around 92 or 93 units, as this is the mean value of the weekly sales data. However, from the scatter plot of sales and price data, you see that different price levels associate with different sales levels. And this suggests that you can use the price to predict sales. To summarize, the linear equations Sales is �a� plus �b� times Price, allows us to predict the effects of a price cut that the store manager did not try before, or to estimate the optimal price to maximize turnover. Now I invite you to consider the following test question. Suppose you are interested to find the price that maximizes turnover, where turnover is defined as the product of Price and Sales. How can you find that price from the relation Sales is �a� + �b� times Price? The answer is as follows. As turnover is Price times Sales, or �a� times P plus �b� times P-squared, the derivative of turnover with respect to price is equal to a + 2 bP. Setting this equal to 0 gives an optimal price of -a / 2b. In our lectures on simple regression, we focus on two variables of interest we denote by y and x, where one variable, x, is thought to be helpful to predict the other, y. This helpful variable x we call the regressor variable or the explanatory factor. And the variable y that we want to predict is called the dependent variable, or the explained variable. The histogram of our sales data may suggest that the sales distribution can be approximated by a normal distribution. A simple way of summarizing the 104 observations on sales is written as shown on the slide. This notation means that the observations of sales are considered to be independent draws from the same Normal distribution, with mean mu and variance sigma squared, abbreviated as NID. Note that we use the Greek letters mu and sigma squared for parameters that we do not know and that we want to estimate from the observed data. The probability distribution of sales is described by just two parameters, the mean and the variance. On this slide you see the graph of a standardized normal distribution with mean 0 and variance 1. And if you wish, you can consult the Building Blocks for further details on the normal distribution. For a normal distribution with mean mu, the best prediction for the next observation on sales is equal to that mean mu. An estimator of the population mean mu is given by the sample mean, where y subscript i denotes the i-th observation on sales. The sample mean is called an unconditional prediction of sales, as it does not depend on any other variable. In many cases, it helps to use additional information to improve the prediction. In our example, price may help to predict sales because lower prices will lead to higher sales. There are four more lectures in this set to come. Lecture 1.2 introduces the simple regression model, and Lecture 1.3 shows how you can find the regression line from the actual data. We will also show how you can evaluate the accuracy of the regression outcomes. In Lecture 1.4 we will examine which statistical assumptions support the use of the simple regression model and what are the consequences when some of these assumptions are not met. The required modifications provide the motivation for the other lectures of this MOOC, that all consist of extensions of the basic ideas of simple regression. And finally, Lecture 1.5 provides some illustrations of the use of simple regression. The modules 2 through 6 of this MOOC provide various extensions, but the simple regression model provides the fundamental basis. And here is another test for you to make, which deals with the variation of points around the imaginary line and the quality of your predictions. The correct answer is diagram B, because this diagram has smallest variation around the imaginary line. For a given price level, the range of potential forecasts for sales is smallest for case B. So here you may expect to make better predictions. Now I invite you to make the training exercise, where you can train yourself with the topics that were treated in this lecture. You can find this exercise on the website. And this concludes our first lecture on Simple Regression.
