Interpretation of positive and negative beta weights in regression equation

I received this elementary question by email:

In a regression equation am I correct
in thinking that if the beta value is
positive the dependent variable has
increased in response to greater use
of the independent variable, and if
negative the dependent variable has
decreased in response to an increase
in the independent variable – similar
to the way you read correlations?

Answer

In explaining the meaning of regression coefficient I found that the following explanation very useful. Suppose we have the regression

$$Y=a+bX$$

Say $X$ changes by $\Delta X$ and $Y$ changes by $\Delta Y$. Since we have the linear relationship we have

$$Y+\Delta Y= a+ b(X+\Delta X)$$

Since $Y=a+bX$ we get that

$$\Delta Y = b \Delta X.$$

Having this is easy to see that if $b$ positive, then positive change in $X$ will result in positive change in $Y$. If $b$ is negative then positive change in $X$ will result in negative change in $Y$.

Note: I treated this question as a pedagogical one, i.e. provide simple explanation.

Note 2: As pointed out by @whuber this explanation has an important assumption that
the relationship holds for all possible values of $X$ and $Y$. In reality this is a very restricting assumption, on the other hand the the explanation is valid for small values of $\Delta X$, since Taylor theorem says that relationships which can be expressed as differentiable functions (and this is a reasonable assumption to make) are linear locally.

Attribution
Source : Link , Question Author : Jeromy Anglim , Answer Author : mpiktas

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