 Dear student, in the previous example, we have seen the one independent variable and we have two dependent variable using the multivariate multiple regression. So, in this example, we have the multivariate multiple regression, sometimes we do not available the raw data. We have calculated values. Now we have calculated values, we do not have the computations, we do not have the values, basically we have the calculation given. Now how do we use the calculation and use the multivariate multiple regression? We will study that in this example. Now this is the example, following information is obtained from a set of 19 observations. So, we have the 19 observation on six variables. These are the six variables, these are the three independent variables and these are the three independent variables. Now the total is the six variables and here is the calculated values of the X transpose X given. Now next is the X transpose Y given. Now we do not have the data given, we have calculated values given. What do we have to do using them? Estimate the multivariate regression model for joint modeling of Y1, Y2, Y3. Find what we have to do? Estimate what we have to do? Multivariate regression use. With that we will get the joint modeling of Y1, Y2, Y3. Simultaneously we will get the equation. We have to consider this. So, this is the example of the multivariate multiple regression. When do we have to use it? When we have computations, i.e. you have results given. We have to use it further and estimate the regression model. Now here is the solution. This is the multivariate regression model. This is the multivariate regression model. Why do you have bold and subcapital? Because Y, X, B, E, we all have matrix system. That is why we called it bold and capital. The estimator of regression matrix B, this is the estimate. B cap which is equals to X transpose X inverse X transpose Y. So, what is given? We have to find the B cap. Given, this is the X prime X. Given, we have to find the inverse. And this is the X transpose Y. You have four cross four matrix. It will be difficult to manually find the inverse of four cross four. Then, we will easily find the inverse of X or R on any of them. Now, we will check on the excel that what is the inverse of four cross four. So, this is the X transpose X and here is the X transpose Y. After multiplication, this is the B cap. So, X transpose X whole inverse X transpose Y. So, what did we do? B cap. The final result is X transpose X whole inverse into X transpose Y. Now, how will we interpret the matrix we have? The multivariate regression model is therefore simultaneously run. Now, simultaneously run, what are you developing? Three univariate regression models are developed. So, the Y1 is the univariate regression model here. This is the B naught minus B1 X1 minus B2 X2 minus B3 X3. This is the Y1. For Y1, it is generated. Now, you have values for model Y2 and these are the values for model Y3. So, what did we do? We ran the multivariate regression models and basically, what are we determined? Three univariate regression model are determined. So, we are taking Y2 from here. This is the B naught. This is the B naught plus B1 X1 plus B2 X2. And the plus B3 X3. This is generated for Y2. And then, we have for Y3. Now, for Y2, this is generated for Y1. So, Y3 equals to this one. B naught plus B1 X1 plus B2 X2 and plus B3 X3. So, what did we say? We can see the multivariate regression model is the collection of three univariate regression models. We ran the multivariate regression model and we determined three univariate multiple regression models. We have the regression model. So, this is the very important example. Basically, what did we do in this example? We had calculated values. We used those calculated values to fit the regression model. And the regression model we have is simultaneously run. We have three univariate regression models generated from the multivariate regression models. So, this is a very interesting example. In daily life, it is very easy to use the multivariate in daily life or in medical life or in social life. Social analysis is used in that. And from there, you can easily convert the multivariate into univariate. So, this is the example of the multivariate multiple regression. So, this is the X-transpose X. Now, we will not manually check this X-transpose X. We will check it on Excel. This is the data you have here. I have entered the X-transpose X in Excel. We have X-transpose X of 4x4 data. Now, I have to find its inverse. So, we have to highlight 4x4 cells. Now, write it down. The equals to M-inwards. Whose M-inwards? This is not the M-inwards. Here is the M-inwards. Whose to find? Highlight these values. Close the bracket. And then, enter shift, control, enter. Now, look at this. This is the inverse of 4x4, previous one. Now, you have the inverse, X-transpose X inverse. Now, we have to multiply this by X-transpose Y. Finally, we will get the value of B-cap. The previous one we have checked here. You will get the result of B-cap. So, on Excel, we have seen a small part. Let's find the inverse of 4x4.