I am just trying to understand how the final prediction is made in multiple regression. Say there are 4 independent variables and one dependent variable. The multiple regression model will first solve the linear relationships between the dependent and each individual variables. Then we have 4 sub linear models don't we? And how is the final value for prediction is calculated from these sub models?
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$\begingroup$ Your question appears to be based on a premise that doesn't hold. It's not actually the case that "multiple regression model will first solve the linear relationships between the dependent and each individual variables". $\endgroup$– Glen_bJul 7, 2017 at 3:44
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2 Answers
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Your question suggests you are not familiar with the concept of multiple regression (not to be confused with multivariate regression). Given that, I will provide you a basic answer and clarification addressing the questions. And, I will also suggests other related topics to study so to get a good educated grasp of the overall subject.
Multiple regression is one single model with one dependent variable Y and several independent variables Xs (multivariate regression is several Ys with several Xs. But, let's keep to the basics for now). A multiple regression is solved using a closed form math formula using Matrix Algebra. The multiple regression uses your entire time series history and can therefore estimate every single point within your time series.
There are no sublinear models. All the variables/regressors are regressed together in one single model as mentioned above.
The final point estimate is not derived any differently than any of the other points within your time series.
To clarify any confusion regarding multiple regression, I recommend related Wikipedia articles on the subject that will confirm all of the above and also give you greater details in the underlying mathematics.
To further understand the topic, I also recommend you study Wikipedia articles on Hold Out samples, cross validation, model testing, etc.
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Calculating predicted values for multiple regression is no different than simple linear regression.
After obtaining your estimated regression equation, you plug the values of the independent variables for each observation. For example, if your estimated equation is $$ \hat{y} = 2x_{1}+3x_{2}+3x_{3}+2x_{4} $$
and suppose the first observation has values (1,0,10,20) for each independent variable, respectively. The predicted value for this observation is calculated as such $$ \hat{y}_{1} = 2(1)+3(0)+3(10)+2(20) = 72 $$
