Random Walks and slope = 1

Random walks are said to have an intercept of 0 and a slope of 1 (p 466). Why can’t a random walk have an intercept other than 0? Why can’t random walks be mean reverting?

This is how i read it. For a random walk the only difference in the dependent variable in the current period and the dependent variable in the previous period is the random error term. In order for this to be possible the intercept would have to be 0 or else the difference between the two dependent variables would be the random error term and the intercept. They can not be mean reverting because the b1 coefficient is always 1, so the mean reverting equation would be b0/(1-1), nothing can be divided by 0 and it is undefined. To be covariance stationary it must have a defined mean reverting level. -Dave

The intercept idea finally clicked when I realized that it’s not the Y intercept that we’re talking about, but rather the vertical “step” from the preceding time increment. All along, I kept wondering why the regression line had to cross the Y axis at 0 when in fact it doesn’t. What still doesn’t click, however, is why a flat line cannot be mean reverting. I get the way it works through the formula (you can’t divide by zero), but I don’t understand it visually. Lastly, stock markets are said to be random walks, yet over time most markets trend upwards. How is this explained?