# Need a Proof for a Complicated Covariance equation

#### sasha 99

##### New Member
Hi everyone
I begin to study time series analysis from a short time , and i found this equation which author did not give a proof for it
so kindly if anyone can simplify this equation to me or give me a more satisfactory explanation for it , it would be appreciated
Here is the equation
https://s32.postimg.org/cw4amnijp/TSA.png

#### Dason

I would just suggest try doing the proof for a simple case first.

$$Cov(c_1X_1 + c_2X_2, d_1Y_1 + d_2Y_2)$$

Just use the definition of covariance and see if you can simplify.

#### sasha 99

##### New Member
Thanks Dason
All what i needed really to make sense for this equation was represent in one of main properties of covariance which is
"Cov( X1 + X2 , Y ) = Cov( X1, Y ) + Cov( X2, Y )"
This is the primary and simple formula which lead me eventually to undertand the product formula

#### sasha 99

##### New Member
About the formula you mentioned above , i gonna through some steps which lead to this result
Cov( c1.X1 + c2.X2 , d1.Y + d2.Y2 ) =

c x d x Cov( X1 + X2, Y1 + Y2 ) =

c x d x ( E [( X1 + X2 - E( X1 + X2 )) x ( Y1 + Y2 - E( Y1 + Y2 ))] =

c x d x ( E [ X1.Y1 - X1.E(Y1) - Y1.E(X1) + E(X1).E(Y1)
X1.Y2 - X1.E(Y2) - Y2.E(X1) + E(X1).E(Y2)
X2.Y1 - X2.E(Y1) - Y1.E(X2) + E(X2).E(Y1)
X2.Y2 - X2.E(Y2) - Y2.E(X2) + E(X2).E(Y2) ] =

c x d x ( COV ( X1,Y1) + COV ( X1,Y2) + COV ( X2,Y1) + COV ( X2,Y2) )
C x D x ∑i=1∑(d=1)(COV ( Xi ,Yd ))