# Time series help, Derive COV in a random walk process

#### eric89

##### New Member
I could use some help deriving the covariance between Y_t and Y_t+h
given the random walk process Y_t = Y_t-1+e_t
And also the correlation between Y_t and Y_t+h

Got stuck..

Also got another problem where i need to state whether pbl_t is correlated with e_(t-1) in this regression q_t= a+Bpbl_t + e_t. I.e. if the pbl_t has any correlation with the lagged value of error term. I am given that the error term e_t is uncorrelated with the present and any past values of pbl_t.

If this is the case, does this violate the Zero conditional mean assumption in the same way as if the explanatory variables are correlated with the error term?

Last edited:

#### BGM

##### TS Contributor
Hi! I see lots of squares � in your post. Maybe you need to check your typed formula again? You may use the $$tag here for LaTeX input.$$

#### eric89

##### New Member
Oops, sorry! Does it look allright now?

#### JesperHP

##### TS Contributor
$$y_t = y_{t-1} + e_t$$
use recursive substitution to derive
$$y_t = y_{0} + \sum_{i=1}^t e_i$$
then calculate
$$cov( y_{0} + \sum_{i=1}^t e_i , y_0 + \sum_{i=1}^{t+h} e_i )=t \sigma^2_e$$

$$var(y_t) = var( y_{0} +\sum_{i=1}^t e_i )= t\sigma^2_e$$

$$var(y_{t+h}) = var( y_{0} + \sum_{i=1}^{t+h} e_i )= (t+h)\sigma^2_e$$

$$corr(y_t,y_{t+h}) = \frac{t\sigma^2_e}{\sqrt{t\sigma^2 (t+h) \sigma^2}} = \frac{t}{ \sqrt{t} \sqrt{t+h}} = \sqrt{\frac{t}{t+h}}$$
If I remember correctly and calculate correctly offcourse