Comparison of β of two linear regressions In my empirical research I use data from two waves of a survey: wave $i$ and wave $j$. My aim is to find out whether there is a difference in the magnitude of the influence of certain explanatory variables on the dependent variable.  To test this question I run a separate regression for each wave ($i$ and $j$):
\begin{align}
Y_i &= β_{1i}X_{1i} + β_{2i}X_{2i}  \tag{1}  \\
Y_j &= β_{1j}X_{1j} + β_{2j}X_{2j}  \tag{2}
\end{align}
Next I test the significance of ($β_{1i} - β_{1j}$) and ($β_{2i} – β_{2j}$).  I would like to make sure that this is the right way to test the significance of the changes in the influence.
 A: This is straightforward using SEM software, like lavaan, to run two separate simultaneous regression models with and without constraints. You can use a Wald test to test whether individual parameters (e.g., the difference in coefficients) is different from 0, or you can do a likelihood ratio test to compare constrained and unconstrained models.
The unconstrained model:
model1 <- '
    Yi ~ ai*X1i + bi*X2i
    Yj ~ aj*X1j + bj*X2j
    Yi~~Yj
    adiff := aj - ai
    bdiff := bj - bi'
fit1 <- sem(model1, data = data)
summary(fit1)

The tests for adiff and bdiff are tests of whether the individual parameter estimates differ between the two models. These tests allow the residual variance of each regression to differ. Note that you need to allow the residual variances to correlate, since they probably do if they are of the same outcome at different times.
Next, the constrained model, which holds the coefficients equal between the two models:
model2 <- '
    Yi ~ a*X1i + b*X2i
    Yj ~ a*X1j + b*X2j
    Yi~~Yj'
fit2 <- sem(model2, data = data)
summary(fit2)

Finally, compare the fit of the two models using a likelihood ratio test. 
lavTestLRT(fit2, fit1)

If significant, there is evidence that at least one of the parameters differs between the models. Otherwise, there is not enough evidence that the parameters differ; unconstraining the coefficients does not add much to the fit of the model.
A: You may run a single regression with indicator variable (say $Z$ being 1 for wave 1 and 0 for wave 2)
and its interactions with $X$'s.
Your model now becomes
$$Y=\beta_1X_1+\beta_2X_2+\beta_3Z+\gamma_1X_1Z+\gamma_2X_2Z$$
Significance of $\gamma_i$ now means that effect of $X_i$ on $Y$ is significantly different for wave 1 and wave 2.
A: The usual interpretation would be - one unit increase in variable X1 leads to Beta1 increase in Y. You could extend this to understand what the difference in impact of X1 is on Y across the two waves, assuming the explanatory and target variables remain the same. However I do not understand how you could possibly calculate the significance of this difference between them.
