Piecewise linear regression with constraint - r

Question

I am trying to do a piecewise linear OLS regression analysis in R, with one breakpoint. I have the following regression formula and restrictions:

Where D is a dummy. I would like to impose a restriction, so that the regression lines are continous (with a break point). The restriction below would work fine.

My question is, how do I formulate that in the lm()-function in R? I have previously tried the "segmented" package, but I need to interpret the intercepts for both lines in a regression summary.

I have provided some data below. The breakpoint here is 0, so d is 1 for x >= 0 .

             x          y d
1    4.3047451 11.2660463 1
2    7.0062423 -3.2897982 1
3    2.7862009 -2.8232636 1
4   -0.8662964  0.4051925 0
5   -0.9553261 -0.9228929 0
6   -1.6626654  3.5044546 0
7    3.4906905  1.4961349 1
8   -0.7072658 -0.2758436 0
9   -7.0054069 -1.3041742 0
10  -2.2510701 -0.1848814 0
11 -13.3617905 -0.2113756 0
12   4.1001251  0.2845748 1
13  -4.6575944 -1.1603290 0
14   5.2243857  3.8324236 1
15   3.5003320 -2.3672985 1
16 -13.2623113 -7.1593177 0
17  -1.7944354 -2.1725478 0
18   0.5885924 -0.2411048 1
19 -19.3744936 -0.1982088 0
20 -17.9876978 -1.5995063 0

Edit:

I have added a graphic representation of what I am trying to perform. It is important that the 2 fitted lines meet at the threshold, and that I can get 4 coefficents. 2 alphas, and 2 betas.

Answer 1

Since the breakpoint is x = 0 we have a = a2 and so:

nls( y ~ (x < 0) * (a + b * x) + (x > 0) * (a + b2 * x), dat,
  start = list(a = 1, b = 1, b2 = 1))

or using lm

lm(y ~ I(x * (x < 0)) + I(x * (x > 0)), dat)

In general if B is the breakpoint:

B <- 0
nls( y ~ (x < B) * (a + b * (x - B)) + (x > B) * (a + b2 * (x - B)), dat,
  start = list(a = 1, b = 1, b2 = 1))

B <- 0
lm(y ~ I((x - B) * (x < B)) + I((x - B) * (x > B)), dat)

Answer 2

This is not an answer but a comment which cannot be edited in the comments section because it requires an image to be understandable.

In fact, I cannot understand your data : When represented on Cartesian graph (below) the points appear very scattered. It doesn't look like a piecewise function. What am I missing ?

By the way, if the points were not too far from a piecewise function made of two inclined segments, there is a very simple method for the fitting. See pages 12-13 in this paper : https://fr.scribd.com/document/380941024/Regression-par-morceaux-Piecewise-Regression-pdf