For the OU process: $dX_i = (\phi - \lambda X_i)dt + \sigma dW_i$
I set σ = 1, φ = 5, λ = 1. Now I want to get four moment conditions for φ and λ (i.e., treat σ as known), where I want to firstly take the moments from the Euler discretization and secondly the exact conditional moment conditions. Hence using the Euler discretization I get:
$g_1(X_i): E[(X_i - X_{i-1} -(\phi - \lambda X_i)\Delta)] $
$g_2(X_i): E[(X_i - X_{i-1} -(\phi - \lambda X_i)\Delta )X_{i-1}] $
$g_3(X_i): E[(X_i - X_{i-1} -(\phi - \lambda X_i)\Delta)^2 -\sigma ^2\Delta ] $
$g_4(X_i): E[(X_i - X_{i-1} -(\phi - \lambda X_i)\Delta )X_{i-1}-\sigma ^2\Delta])X_{i-1} $ ]
For the exact solution I dont really now how to proceed. I think I have to use the solution of the OU process with conditional expectation:
$E[X_t | X_s] = X_se^{-\lambda(t-s)}+\phi/\lambda(1-e^{-\lambda(t-s)})$
and conditional variance: $Var[X_t|X_s]=\sigma^2/2\lambda(1-e^{-2\lambda(t-s)})$, but I dont know how to express this now in terms of moment conditions.
Also I would like to implement these GMM estimators for the given parameter values. Would I just need to setup the matrix with these conditions and then use the gmm function in R?