- #1

member 428835

Given random time series data ##y_i##, we assume the data follows a EWMA (exponential weighted moving average) model: ##\sigma_t^2 = \lambda\sigma_{t-1}^2 + (1-\lambda)y_{t-1}^2## for ##t > 250##, where ##\sigma_t## is the standard deviation, and ##\sigma_{M=250}^2 = \sum_{i=1}^{250}y_i^2/250## to initialize. How would we use maximum likelihood to estimate ##\lambda##?

In general, it seems to use the principal we first choose a distribution ##P(y_i)## the data likely came from ( like a Bernoulli variable maybe for flipping a coin and estimating probability of heads ##p##, or normal distribution if we've been given heights of people as a sample and want to estimate the mean, standard deviation). Next, since the data are i.i.d. (we assume this is true) we optimize ##\Pi_i P(y_i)## with respect to the variable we seek (##p## or ##\mu## in the previous examples, in this question should be ##\lambda##). I'm just confused how the assumed model with ##\sigma## plays a role. Any help is greatly appreciated.