Linear Regression with OLS/Lasso/Ridge

You are given a design matrix X ∈ ℝ^{n×p} and a response vector y ∈ ℝ^{n}. Implement three estimators for the linear model y = Xβ + ε and compare their behaviour under multicollinearity.

1. Ordinary Least Squares (OLS): β̂ = (XᵀX)⁻¹Xᵀy.
2. Ridge regression (L2): β̂ = (XᵀX + λI)⁻¹Xᵀy with λ > 0.
3. Lasso regression (L1): β̂ = argmin_β {‖y − Xβ‖₂² + λ‖β‖₁} with λ > 0.

Your task is to write a single Python module that

• fits each model to the provided training data,
• returns the estimated coefficients β̂,
• computes the mean-squared error on a held-out test set,
• reports the number of non-zero coefficients for each method,
• and plots the coefficient paths (β̂ vs λ) for Ridge and Lasso over the grid λ ∈ [10⁻², 10⁻¹, 1, 10, 10²].

Use only NumPy, SciPy, and Matplotlib; scikit-learn is not allowed. Handle singular XᵀX gracefully for OLS via the pseudo-inverse.