Explain the difference between multiple regression and multivariate regression, with minimal use of symbols/math Are multiple and multivariate regression really different? What is a variate anyways?
 A: Here are two closely related examples which illustrate the ideas. The examples are somewhat US centric but the ideas can be extrapolated to other countries.
Example 1
Suppose that a university wishes to refine its admission criteria so that they admit 'better' students. Also, suppose that a student's grade Point Average (GPA) is what the university wishes to use as a performance metric for students. They have several criteria in mind such as high school GPA (HSGPA), SAT scores (SAT), Gender etc and would like to know which one of these criteria matter as far as GPA is concerned.
Solution: Multiple Regression
In the above context, there is one dependent variable (GPA) and you have multiple independent variables (HSGPA, SAT, Gender etc). You want to find out which one of the independent variables are good predictors for your dependent variable. You would use multiple regression to make this assessment.
Example 2
Instead of the above situation, suppose the admissions office wants to track student performance across time and wishes to determine which one of their criteria drives student performance across time. In other words, they have GPA scores for the four years that a student stays in school (say, GPA1, GPA2, GPA3, GPA4) and they want to know which one of the independent variables predict GPA scores better on a year-by-year basis. The admissions office hopes to find that the same independent variables predict performance across all four years so that their choice of admissions criteria ensures that student performance is consistently high across all four years.
Solution: Multivariate Regression
In example 2, we have multiple dependent variables (i.e., GPA1, GPA2, GPA3, GPA4) and multiple independent variables. In such a situation, you would use multivariate regression.
A: Very quickly, I would say: 'multiple' applies to the number of predictors that enter the model (or equivalently the design matrix) with a single outcome (Y response), while 'multivariate' refers to a matrix of response vectors. Cannot remember the author who starts its introductory section on multivariate modeling with that consideration, but I think it is Brian Everitt in his textbook An R and S-Plus Companion to Multivariate Analysis. For a thorough discussion about this, I would suggest to look at his latest book, Multivariable Modeling and Multivariate Analysis for the Behavioral Sciences.
For 'variate', I would say this is a common way to refer to any random variable that follows a known or hypothesized distribution, e.g. we speak of gaussian variates $X_i$ as a series of observations drawn from a normal distribution (with parameters $\mu$ and $\sigma^2$). In probabilistic terms, we said that these are some random realizations of X, with mathematical expectation $\mu$, and about 95% of them are expected to lie on the range $[\mu-2\sigma;\mu+2\sigma]$ .
A: Simple regression pertains to one dependent variable ($y$) and one independent variable ($x$): $y = f(x)$
Multiple regression (aka multivariable regression) pertains to one dependent variable and multiple independent variables: $y = f(x_1, x_2, ..., x_n)$
Multivariate regression pertains to multiple dependent variables and multiple independent variables: $y_1, y_2, ..., y_m = f(x_1, x_2, ..., x_n)$. You may encounter problems where both the dependent and independent variables are arranged as matrices of variables (e.g. $y_{11}, y_{12}, ...$ and $x_{11}, x_{12}, ...$), so the expression may be written as $Y = f(X)$, where capital letters indicate matrices.
Further reading:


*

*"R Cookbook" by P. Teetor, O'Reilly publisher, 2011, Chapter 11 on "Linear Regression and ANOVA".

*Quora question "What is the difference between a multiple linear regression and a multivariate regression?"

*Mathworks (Matlab) tutorial on linear regression.

A: In multivariate regression there are more than one dependent variable with different variances (or distributions). The predictor variables may be more than one or multiple. So it is may be a multiple regression with a matrix of dependent variables, i. e. multiple variances. 
But when we say multiple regression, we mean only one dependent variable with a single distribution or variance. The predictor variables are more than one. 
To summarise multiple refers to more than one predictor variables but multivariate refers to more than one dependent variables. 
A: I think the key insight (and differentiator) here aside from the number of variables on either side of the equation is that for the case of multivariate regression, the goal is to utilize the fact that there is (generally) correlation between response variables (or outcomes). For example, in a medical trial, predictors might be weight, age, and race, and outcome variables are blood pressure and cholesterol. We could, in theory, create two "multiple regression" models, one regressing blood pressure on weight, age, and race, and a second model regressing cholesterol on those same factors. However, alternatively, we could create a single multivariate regression model that predicts both blood pressure and cholesterol simultaneously based on the three predictor variables. The idea being that the multivariate regression model may be better (more predictive) to the extent that it can learn more from the correlation between blood pressure and cholesterol in patients.
A: There ain’t no difference between multiple regression and multivariate regression in that, they both constitute a system with 2 or more independent variables and 1 or more dependent variables. As long as the outcome doesn’t depend on lag obs or a single predictor, it’s called multiple or multivariate regression otherwise it is termed univariate regression.
