# Maximum likelihood to fit a parameter of this model

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• member 428835

#### member 428835

Hi PF!

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.

The squaring just adds unnecessary superscripts for this exercise so let's write ##S_i## for ##\sigma^2_i## and ##X_i## for ##y_i^2##.

Typically we assume the ##X_i## are iid. Say the distribution of ##X_i## has parameters ##\mathbf \beta=\langle \beta_1,...,\beta_K\rangle##, and the probability density function of ##X_i## is ##f_{\mathbf \beta}##. We need to estimate ##\mathbf\beta## and ##\lambda## given observations ##s_1, ...,, s_n## for the random variables ##S_1, ..., S_N##.

Given the equations
$$S_t = \lambda S_{t-1} + (1-\lambda)Y_{t-1}$$
for ##t=2,...N##
and the missing equation ##S_1=X_1##
we can write the realized values ##x_1,..., x_N## of the random variables ##X_1,..., X_N## in terms of just ##\lambda## by inserting the observed values of ##S_1, ..., S_N##. Write these as ##x_1(\lambda),..., X_N(\lambda)## to emphasise this dependence.

The likelihood of the observed data given ##\mathbf \beta,\lambda## is
$$\mathscr L (\mathbf \beta,\lambda)= \prod_{i=1}^N f_{\mathbf\beta}(x_i(\lambda))$$

This expression has ##K+1## unknowns: ##\beta_1, ..., \beta_K, \lambda##. We partially differentiate it wrt each of those unknowns in turn and set it equal to zero, to get ##K+1## equations, the same number as we have unknowns. Solving those equations leads to the ML estimators of those unknowns.

Note how we needed the density function of ##X_i## to form the expression for ##\mathscr L##.