Karl Ho
School of Economic, Political and Policy Sciences
University of Texas at Dallas
\(C_1(X) = I(X < 35), C_2(X) = I(35 <= X < 50),...,C_3(X) = I(X>= 65)\)
Smooth, local but no continuity
Top Left: The cubic polynomials unconstrained.
Top Right: The cubic polynomials constrained to be continuous at age=50. Bottom Left: Cubic polynomials constrained to be continuous, and to have continuous first and second derivatives. Bottom Right: A linear spline is shown, constrained to be continuous.
Knot at 50
$$y_{i}=\beta_0+\beta_1b_1(x_i)+\beta_2b_2(x_i)+ ... +\beta_{K+1}b_{K+1}(x_i) + \epsilon_i$$
Truncated function
Starting at 0 for continuity
Starting at 0 for continuity
$$y_{i}=\beta_0+\beta_1b_1(x_i)+\beta_2b_2(x_i)+ ... +\beta_{K+3}b_{K+3}(x_i) + \epsilon_i$$
Truncated power function
Adding the last term in the cubic polynomial will lead to a discontinuity in only the third derivative at \(\xi\); the function will remain continuous, with continuous first and second derivatives, at each of the knots.
Natural cubic spline is better!