# 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?

Thanks in advance 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