EWMA formula

[kevolution]

I noticed in the Ch. 23 Hull video, it shows the EWMA formula in both extended form and recursive form. On the recursive formula, sigma(n)^2 = lambda * sigma(n-1)^2 + (1 - lambda)*u(n-1)^2; however, the extended formula shows that the formula consists of (1-lambda)*lambda0*u(n-1)^2 + (1-lambda)*lambda1*u(n-2)^2... etc

so I'm wondering what the difference is between these two equations and why one's coefficient is (1-lambda) while the other is lambda

 

[ShaktiRathore]

Hi,
(1-lambda)*lambda^0*u(n-1)^2 + (1-lambda)*lambda^1*u(n-2)^2...
=(1-lambda)*u(n-1)^2 + lambda^1*((1-lambda)*lambda^0*u(n-2)^2+.....)
=(1-lambda)*u(n-1)^2 + lambda*(sigma(n-1)^2 ) is other form (since sigma(n-1)^2 =(1-lambda)*lambda^0*u(n-2)^2+.....)
thanks

 

[kevolution]

Thanks this makes sense.

 

[kevolution]

The GARCH example in the same video, I seem to get a different answer than 1.43%... I get 0.0205% = .00798^2 + .00014082