Least Square Estimator Derivation for 2-Dimensional Stochastic Process I am trying to work through an example in this paper Least squares estimators for discretely observed
stochastic processes
The authors give the following
$$
 \Psi_{n,\epsilon}(\theta) = \sum_{k=1}^n \frac{\lvert X_{t_k} - X_{t_{k-1}}-b(X_{t_{k-1}},\theta)\Delta_{t_{k-1}}\rvert^2}{\epsilon^2\Delta_{t_{k-1}}}
$$
of which minimizing gives the Least Square Estimator.
The example I am working through is as follows:

The authors state after some basic calculations they achieve the LSE; however, it doesn't seem so basic. If I try to write out $\Psi_{n,\epsilon}(\theta)$ in this case, then it becomes very messy quickly. Also, the matrix $\Lambda_n$ is not invertible, nor is it clear to me where they got that from. So I must be missing something as I assume the example is indeed correct.
Would appreciate if someone would enlighten me.
 A: Define $B^T=[C,A],\,\tilde{y}=[1,y^{(1)},y^{(2)}]^T$. Note, for $x,y \in \mathbb{R}^2$
$$\begin{aligned}|x-(C+Ay)n^{-1}|^2&=|x-B^T\tilde{y}n^{-1}|^2=\\
&=x^Tx-2x^TB^Tyn^{-1}+\tilde{y}^TBB^T\tilde{y}n^{-2}\end{aligned}$$
and by linearity
$$\varepsilon^{-2}n\sum_{k\leq n}|x_k-(C+Ay_k)n^{-1}|^2=\varepsilon^{-2}n\sum_{k\leq n}x_k^Tx_k-2\varepsilon^{-2}\sum_{k\leq n}x_k^TB^T\tilde{y}_k+\varepsilon^{-2}n^{-1}\sum_{k\leq n}\tilde{y}^T_kBB^T\tilde{y}_k$$
take the derivative wrt to $B$ and set to $0$:
$$-2\sum_{k\leq n}\tilde{y}_kx_k^T+2n^{-1}\bigg(\sum_{k\leq n}\tilde{y}_k\tilde{y}^T_k\bigg)B=0\implies B=\bigg(\sum_{k\leq n}\tilde{y}_k\tilde{y}_k^T\bigg)^{-1}\bigg(n\sum_{k\leq n}\tilde{y}_kx_k^T\bigg)$$
Now note
$$\tilde{y}_k\tilde{y}_k^T=\begin{bmatrix}1&y^{(1)}_k&y^{(2)}_k\\
y^{(1)}_k&(y_k^{(1)})^2&y_k^{(1)}y_k^{(2)}\\
y^{(2)}_k&y_k^{(1)}y_k^{(2)}&(y_k^{(2)})^2
\end{bmatrix},\,\tilde{y}_kx_k^T=
\begin{bmatrix}x^{(1)}_k&x^{(2)}_k\\
x^{(1)}_ky^{(1)}_k&x^{(2)}_ky_k^{(1)}\\
x^{(1)}_ky^{(2)}_k&x^{(2)}_ky_k^{(2)}
\end{bmatrix}$$
