Broken Line Regression

Question

$X = $Lot & $Y = $Cost

Give a broken line linear model with a breakpoint at $250$: $$Y = B_0 + B_1X_1 + B_2X_2 + B_3X_3 + e$$ where $X_2 = 0$ or $1$ depending on whether the lot size is $\geq 250$ or $< 250$ and $X_3 = X_1\cdot X_2.$

Which hypothesis statement is equivalent to the statement: The two regression lines have the same intercept term? $$H_0 : B_0 = 0\\ H_0 : B_1 = 0\\ H_0 : B_2 = 0\\ H_0 : B_3 = 0$$

$B_0$ is obviously the intercept for simple linear regression models, however, the broken line phrasing is causing me to be confused on this. My initial instinct was to assume $B_0$ hypothesis was appropriate, but now I'm wondering if $B_2 = 0$ makes more sense.

Any assistance or starting point would be very beneficial as I need to explain through the reasoning algebraically too !

Answer 1

Presumably $X_1$ is the lot size

When $X_1 < 250$ you have $X_2=0$ and $X_3=X_1X_2=0$ so

• $Y = B_0+B_1X_1$ with intercept $B_0$

When $X_1 \ge 250$ you have $X_2=1$ and $X_3=X_1X_2=X_1$ so

• $Y = (B_0+B_2)+ (B_1+B_3)X_1$ with intercept $B_0+B_2$

so the intercepts are equal when $B_2=0$