# Test if regression coefficients are equal between periods of time

I have multiple observations per year and I was asked to estimate an individual regression line for each year within a single model.

My data has three years (1979, 1980 and 1981) and two independent variables ($$X_1$$ and $$X_2$$).

My model looks as follows: $$y_i = D_{79,i} \big( \beta_0 + \beta_1 X_{1,i} + \beta_2 X_{2,i} \big) + \\ D_{80,i} \big(\beta_3 + \beta_4 X_{1,i} + \beta_5 X_{2,i} \big) + \\ D_{81,i} \big(\beta_6 + \beta_7 X_{1,i} + \beta_8 X_{2,i} \big) + u_i$$

Where $$D_{y,i}$$ is an indicator for year $$y$$.

In essence, this model fits one regression line per year to each group of observations.

How can I carry out an F-test to test if all the $$\beta$$ coefficients associated with $$X_1$$ and $$X_2$$ are the same between each year?

That is, test if $$\{\beta_1 = \beta_4 = \beta_5\}$$ and $$\{\beta_2 = \beta_5 = \beta_6\}$$ simultaneously.

This seems to be equivalent to the following linear regression model in R formula syntax:

modelFull <- lm(y ~ (X_1 + X_2) * Yr)


where Yr is a 3-level categorical predictor representing the 3 different years.* That allows all coefficients, including the intercept implicit in that syntax, to depend on the value of Yr.

The simplest overall F-test would be to test that model against a model that omits Yr:

modelSimple <- lm(y ~ X_1 + X_2)


which returns a single intercept and single coefficients for each of X_1 and X_2. In R, the anova() function uses an F-test by default to compare 2 linear regression models.

*The way you wrote the model doesn't allow you to estimate values of all of the specified parameters independently. You have listed 13 parameters, but only 9 coefficients can be estimated uniquely: 3 individual coefficients including the intercept for each year, times 3 years.

• Thanks! It makes sense that I was asked to estimate the model without year-specific terms before doing this. Just to make things clear, the terms $D_{y,i}$ are not parameters. They are dummy variables that take the value of 1 when the year equals $y$. Nov 28, 2021 at 21:56