How to determine if predictor in piece-wise linear regression is significant?

Question

I saw a comment by mpiktas that "The sum of two independent t-distributed random variables is not t-distributed". Is it then of any known distribution?

Actually, I am using piece-wise linear regression.

$E(y) = \beta_0 + \beta_1x + \beta_2(x - break)\cdot d$

where

break = value of predictor x at the breakpoint 
d = dummy variable = 1, if x > break; 0, otherwise

Effectively, the equation means:

$E(y) = \beta_0 + \beta_1x$ if x > break

$E(y) = (\beta_0 – \beta_2 \cdot break) + (\beta_1+\beta_2)x$ otherwise

And I want to know whether $x$ is a significant predictor. Couldn't find anything similar on the internet. Thus, I thought of:

1. calculating the p-value for $(\beta_1+\beta_2)$ to determine if $x$ is a significant predictor on the right of breakpoint (i.e. when x > break); and
2. using p-value for $\beta_1$ to determine if $x$ is a significant predictor on the left of breakpoint (i.e. when x =< break).

Answer 1

These are called piecewise linear splines, you should be able to find some references on the internet also regarding different possible parametrizations (the coefficient of the second spline may represent the change in the slope from the preceding interval or it may represent the slope for the second interval).

Anyway, if you want to test against the null $H_0: {\beta}_{spline1} + {\beta}_{spline2} = 0$ you can construct the significance test by hand (or, if you're using Stata, using the test postestimation command).

Fit your unrestricted model and then fit your restricted model under the constraint $\beta_{spline1} + \beta_{spline2} = 0$. At this point you can construct a $F$ test

$$ F_{obs} = \frac{(SSE_{restricted}-SSE_{unrestricted})}{SSE_{unrestricted}/(n-k_{unrestricted}-1)} $$

which under the null is distributed as a Snedecor's $F$ r.v. with $1$ and $n-k_{unrestricted}-1$ degrees of freedom.

($n =$ # observations, $k =$ # predictors)