Difference between the effect and the contribution of a regressor

Question

I have a very basic question about the difference between effect of a regressor and its contribution. Consider the following simple case. In economics, Gross Domestic product is defined as the sum of consumption, investment, government expenditure and net exports. Consider that we have the following time series of observations: $C_{t},I_{t},G_{t},X_{t}-M_{t}$

Now, $Y_{t}$ is defined as:$$Y_{t}=C_{t}+I_{t}+G_{t}+X_{t}-M_{t}$$

If we were to run a regression of $Y_{t}$ on the RHS regressors, we would get an $R^{2}$ of 1 and \beta of 1 as well. This is because this model is deterministic. My question is, although the effect of these regressors is all the same, how can we judge the contribution of each of these to the regressand? Is partial $R^{2}$ a good way to do this? Also, the contribution will be different for each observation. Would the partial $R^{2}$ be a measure of average contribution?

Answer 1

I think I figured the answer out. Ultimately, I am trying to compare the partial $R^{2}$ to the coefficient. They are in some ways related. Consider we have a model:$$y=x\beta+\epsilon$$ where $x$ is a single variable. Now, we know that:$$\beta=\frac{cov(x,y)}{var(x)}$$

Moreover, we know that $$R^{2}=\rho^{2}$$ Also,$$\rho_{xy}=\frac{cov(x,y)}{\sigma_{x}\sigma_{y}}$$

This implies that $$\rho_{xy}=\beta\frac{\sigma_{x}}{\sigma_{y}}$$ which implies that:$$R^{2}=\beta^{2}\frac{var(x)}{var(y)}$$ This is the difference between “importance” and “significance.” We can have a completely/highly significant regressor, but if its variation is not “large” compared to the “regressand”, it is not important. We do however, identify the partial effect of x consistely, given appropriate exogeneity assumptions. This is why we make a distinction between things that are economically significant and things that are statistically significant.