Regressor contribution in OLS regression Assume I have the following model, estimated using OLS:
$Y_{it}=β0+β1∗X1_{it}+β2∗X2_{it}+β3∗X3_{it}+ϵ_{it}$
I know that some methods exist to compute the relative contribution of each variable to the overall goodness of fit (Shapley decomposition for instance). But is there any easy way to measure a time-varying contribution? I.e. the evolution over time of the contribution of each factor?
Probably silly question: what does $(β1∗X1_{it})/Y_{it}$ represent? Can this represent some kind of contribution of $X_1$ to $Y$?
Many thanks
 A: If your dataset is a time-series dataset, i.e. the data has been collected at varying points in time, then the regression model will likely suffer from autocorrelation, where there is a correlation between the observations at time t and t-1, etc.
The reason I use the term "suffer" is because autocorrelation is a violation of Ordinary Least Squares regression as being the Best Linear Unbiased Estimator (BLUE).
According to Studenmund: Using Econometrics, A Practical Guide (2014), serial correlation causes the standard errors of the OLS estimates to be biased, which leads to unreliable hypothesis testing, i.e. we are likely to make a Type 1 or Type 2 error.
In this regard, first-differencing is often a common procedure applied to ensure no serial correlation among the variables, while still allowing for an estimation of the impact of one variable on another, i.e. what is the impact of a unit change in X1 on Y?
Therefore, the purpose of this method is to isolate the impact of time rather than measure a time-varying contribution per se - as serial correlation is a violation of OLS.
That said, it is unreasonable to expect that the coefficients will be constant over time. One potential solution is to partition the data and run first-differenced regressions across these different time periods and observe any potential differences.
