Today I explore argmax and argmin function defined in Base package. armax returns a maximal index, while argmin returns a minimal index. A = [1 2; 3 4] argmax(A) Output: CartesianIndex(2, 2) B=[-1 2; 10 -10] argmax(B) Output: CartesianIndex(2, 1) argmin(B) Output: CartesianIndex(2, 2) argmax can have a function as the first argument (say f) and the domain of the function (say x) as the second argument and it will still find a value of x that maximizes f(x) ....
Suppose we have following AR(1) model (autoregressive model of order 1) defined by: $$ X_{t+1}=aX_t + b+ cW_{t+1} $$ where \(X_0\) and \(W_{t}\) are independent, [a,b,c] are scalar-valued parameters, the state space is \(\mathbb{R}\), the process \(W_{t}\) is independently distributed and standard normal, and initial condition \(X_0\) is drawn from the normal distribution of mean \(\mu_0\) and standard error of \(\nu_0\). Iterating backward from time \(t\), we have $$ X_t = a^tX_0 + b \sum_{j=0}^{t-1}a^j + c\sum_{j=0}^{t-1}a^jW_{t-j}, $$ which implies that \(X_t\) depends on parameter [a,b,c], initial condition \(X_0\), and the sequence of shocks from period 1 throughout t represented by \(W_1, \cdots, W_t\)....