Hi Guys, am doing this time series question. im not sure how we are supposed to know that the conditional distirbution is N(alphaX_t-1, omega^2). is there a rule i dont know? Thanks, Molly
Hi Molly The time series equation is: \( X_t = \alpha X_{t-1} + e_t \) There are two random components to the construction of \( X_t \), there is \( X_{t-1} \) and \( e_t \). If we know the value of \( X_{t-1} \) then the only random part left is the \( e_t \), which we're told are \( N(0, \sigma^2) \) random variables. So, for example: \( X_t | X_{t-1} = 5 \) follows the \( N(\alpha * 5, \sigma^2) \) distribution. This comes from adding the constant \( \alpha * 5 \) to a \( N(0, \sigma^2) \) random variable. More generally: \( X_t | X_{t-1} = x_{t-1} \) follows the \( N(\alpha * x_{t-1}, \sigma^2) \) distribution. Leaving \( X_{t-1} \) in random variable form, we can also write: \( X_t | X_{t-1} \) follows the \( N(\alpha * X_{t-1}, \sigma^2) \) distribution. Hope this helps! Andy
Hi Andy, Thank you so much. Using your notes i was able to complete the question, so thank you does this mean that we are only able to use maximum likelihood estimation for lags of 1?
