Hello everyone. In this video, you will learn how to include more variables into the model, and build a multiple regression model for trend, price and other factors. Let's first look at the retail price data. Since the introduction of the product in January, 2011, the retail price was kept at $949.98 per unit, until September, 2012. Now in September 2012, the retail price was dropped to $799.95 per unit, and stayed on the same since then. So we create another column next to the time variable, for the price data. We also speculate that the home sales may affect the sales of durable cookware because families may purchase new cookware when they move into new homes. Thus, we collected monthly home sales data in the US, as indicated by the blue triangles in this scatter plot. We create another column for the total home sales in the US, say column E. Now we're ready to build the multiple regression model for the sold units, based on three independent variables. They are; time in month, 10-Piece set sale price, total home sales. Following the same procedure as in the simple linear regression, we bring up a dialog box of regression. We will first select the input Y range, to be in the column of the sold units. Please include the first row, which is the label, then we select all the independent variables in one block for the input X range. Now make sure check the box of labels, and specify where the output range should be, and from the residuals box, check residuals, residual plots, line fit plots, and then "Okay". Here are the outputs, in the Summary Output we found that R-Square increased significantly from 43.9 percent to 84.8 percent. The Coefficients table shows the estimated model parameters, and we can see that both the time in month and sale price variables are significant at the 0.1 significance level because their P-values are less than 0.1. The coefficients mean that on average the sold units increased by 12.48 units per month, and for every dollar of price drop, the monthly sold units increased by 6.27 units. However, the home sales variable is not significant, with P-value much greater than 0.1. Its coefficient is also negative, which does not make sense because we expected a higher home sales will lead to more cookware sold. Based on the Coefficients table, we can determine the regression equation to be Y hat, which is the predicted demand, equals to 6,541.28, which is the intercept, plus 12.48 times X1, which is a time in month, minus 6.27 times X2, which is the retail price, and minus 69.5 times X3, which is the home sales. The line fit plot with respect to time is shown in the figure to your right. You can see is that, with the price variable included, the model properly captures the high demand towards the end of the time horizon. For multiple regression, the best way to visualize the fit of the model is to do a scatter plot between the predicted sold units, and the actual sold units. If the model is sufficiently accurate, then all the dots should be on this 45-degree line. As we can see from this figure, the model is a nice fit to the data but still has some errors. To validate the model and to further improve the model, we plot out the residuals along the time. As we can see the equal variance condition is met in this model. However, the residuals may still have a periodical pattern. That is, they are not independent. To better detect the periodical pattern, we generate a line plot for the residuals. Where the blue dots are the residuals of 2011 data and the red dots are the residuals of 2012 data. From this figure, we can see that the residuals do have a periodical or seasonal pattern over a year. In summary, our multiple regression model shows that the trend and price variables are significant but the home sales is not. Together these variables can explain about 84.8 percent of the variations in the demand data. Our residual analysis also shows that, the residuals have a periodic pattern over a year. To further improve the model, we plan to include seasonality, such as holidays and wedding seasons into the model.