Rewriting estimation equations

With regards to my previous post on a volatility estimator(Rolling variance estimator), based on the risk metrics estimator by JP morgan

I have identified the problem to be that the variables should be raised to the power E in order to work. The variables are used during the stochastic simulation.
But what would be the proper way of writing this, which combination of variables should be written like exp(…)?
The goal is to rewrite:

mean_y_1(t) =alpha_1*mean_y(t-1) + (1-alpha_1)y(t-1)
mean_y_2(t) =alpha_2
mean_y(t-1) + (1-alpha_2)*y(t-1)

Var_y_1(t) = alpha_1Var_y_1(t-1) + (1-alpha_1)(y-mean_y_1(t))(y-mean_y_1(t-1))
Var_y_2(t) = alpha_2
Var_y_2(t-1) + (1-alpha_2)(y-mean_y_2(t))(y-mean_y_2(t-1))

Cov_y_1_2(t) = (1-alpha_3)(Cov_y_1_2(t-1) +alpha_3(y-mean_y_1(t))*(y-mean_y_2(t)))

Cor_y(t) =Cov_y_1_2(t)/(sqrt(var_y_1)*sqrt(var_y_2))

to estimators where the variables are expressed in terms of exponentials, but how should this be done?

So, should
mean_y(t) =alpha*mean_y(t-1) + (1-alpha)y(t-1)
Be expressed as:
exp(mean_y(t)) =alpha
exp(mean_y(t-1)) + (1-alpha)*exp(y(t-1))

Var_y_1(t) = alpha_1Var_y_1(t-1) + (1-alpha_1)(y-mean_y_1(t))(y-mean_y_1(t-1))
Be expressed as:
exp(Var_y_1(t)) = alpha_1
exp(Var_y_1(t-1)) + (1-alpha_1)(exp(y)-exp(mean_y_1(t)))(exp(y)-exp(mean_y_1(t-1)))
exp(Var_y_1(t)) = alpha_1exp(Var_y_1(t-1)) + (1-alpha_1)(exp(y-mean_y_1(t))*(y-mean_y_1(t-1)))
or something else?

And should the correlation coefficient be written like:

exp(Cov_y_1_2(t)) = (1-alpha_3)(exp(Cov_y_1_2(t-1)) +alpha_3(exp(y)-exp(mean_y_1(t)))(exp(y)-exp(mean_y_2(t))))
exp(Cov_y_1_2(t)) = (1-alpha_3)exp(Cov_y_1_2(t-1) +alpha_3exp(y-mean_y_1(t))

or maybe even(although this is probably incorrect)
exp(Cov_y_1_2(t)) = (1-alpha_3)(exp(Cov_y_1_2(t-1)) +alpha_3(exp(y-mean_y_1(t)))*(exp(y-mean_y_2(t))))

and should the correlation coefficient be written like:

Cor_y(t) =Cov_y_1_2(t)/(sqrt(var_y_1)*sqrt(var_y_2))


exp(Cor_y(t) ) =exp(Cov_y_1_2(t))/(sqrt(exp(var_y_1))*sqrt(exp(var_y_2)))

and how would this impact the choice of alpha? since now the alpha’s originally were around 1/10?

Any help would be greatly appreciated!

Sorry, but I don’t really understand what you are trying to do here. In the original post, you had a linear model that be definition is certainty equivalent and tried to compute variances that are not time-varying in this case.

Dear Dr. Pfeifer,

First of all, thank you for your reply.
To give some context, we are trying to model the financial markets reaction to shocks, and model the way certain option are traded.
In order to do so, we employ simple methods/pricing tactics like black sholes and therefore need a realistic measures of risk, including volatility by using the RiskMetrics estimator popularized by JP morgan.
Since whether being outside of the steady state is not something that can directly be measured, volatility is a used as a measure of instability.
When analysing for example the Smets/Wouters model, it can be shown variance is indeed time dependant, which we confirmed by GARCH estimation. And since a riskmetrics estimator for volatility, has its own imperfections(and trading accordingly would induce its own feedback mechanisms, which are suspected to be a component of the 07/08 crisis), I would be surprised all this imperfection would be decomposed away.
Also it was my understanding, when solving the stochastic model to a higher order, this certainty equivalence should not present? Or does this only work for third order stochastic non-linear models?

Anyhow, I can understand how certainty equivalence would present itself after decomposition, but before, this should isn’t that clear since the estimators are imperfect and there are many different sources of endogenous volatility as can be shown by factor analysis.

And when I wrote the estimator in terms of exponents, it gives a rather striking graph of decaying volatility in the smets/wouters model.
So now I am left to wonder what it was what we were looking at?
We wrote the exponential weighted moving average variance estimator (where the mean is calculated with by EWMA as well, with its mean as an initial value).
var(t)=(1-a)var(t-1) +a*(y-mean(y))*(y-mean(y-1))

as such
exp(Var_y_1(t)) = alpha_1exp(Var_y_1(t-1)) + (1-alpha_1)(exp(y-mean_y_1(t))*(y-mean_y_1(t-1)))

And I am left to wonder whether writing the estimator in terms of exponents is correct?

since modelling agents who make decisions based of second/third/etc moments would be pretty interesting (and I guess even vitally important when agents are basing decisions on forecasts instead of only rational expectations).



My point was a simple technical one. If your underlying equilibrium system that governs the endogenous variables is linear, the variance cannot be time-varying due to linearity. In this case, however you implement your variance estimator, you will get nonsense. You need to have a nonlinear model that is solved to at least third order (depending on the structure, second order might sometimes be sufficient)