 # random walk

im am having some trouble wrapping my head around this. i get the formula but dont understand the concept. how is it different from a unit root. the cfa material describes it in a very technical way. can anyone simplify the concepts here in plain language?

stuff that happened last time is gonna happen again at least thats my take

The best way to think of random walk with no drift is the best predictor of X at time = t is X at time = t-1 or, the best predictor of next period’s value is this period. If you add a drift, the best predictor of next period’s value is this period’s value plus whatever the drift is.

Consider this. You have an apple and an apple tree. The apple comes from the apple tree, not vice versa. The apple is the unit root, and the apple tree is a random walk. Now, for a more fun explanation-------> My understanding is that a random walk has a unit root. The unit root simply means that b1 = 1. This means that X(t) will be what X(t-1) was plus some error term. This is where it can get confusing because the error term is supposed to be zero, which would imply that X(t) = X(t-1) = X(t-n)… That said, a unit root is not a random walk. A random walk (with and without drift) both have a unit root. The unit root simply means that b1 = 1. _____ For more info, in a random walk, data is noncovariance stationary. You must first difference the data (which means you’ll have n-1 observations, remember). First differencing is easy, it’s just the difference between the 1 & 2 observation, 2 & 3 observation, and so on. Once first differenced, T-test the intercept, lagged terms, & autocorrelation of the residuals. If any are statistically significantly different from zero, you have a random walk! (FUN, eh?!) ----- Now, with a unit root, I think a key thing to remember is that ONE time series is the independent variable and ANOTHER COMPLETELY DIFFERENT times series is the dependent variable. This can be different from a random walk, which is just an AR model (X(t-n) are lagged values of Xt, the dependent variable). That said, let’s say you’re using the log of home sales data (you gotta transform sales data, remember, to make it covariance stationary) to explain consumer sentiment (totally unrelated probably, I know, I know). Let’s say a unit root exists. That means for a 1 unit change in home sales, you perceive a 1 unit change in consumer sentiment. If so, then the data is not covariance stationary, and a unit root may exist. You use Dickey-Fuller (Engle-Granger) T-stats to test for unit root. Then, you go on to testing for cointegration, but thats above my paygrade.

I think they are very similar topics for L2. Unit roots and random walks are a one-to-one mapping when the order of the autoregression is 1 (i.e. when there is one slope coefficient). A random walk is an AR(1) time series where the slope is 1. A unit root is a property of a time series that causes non-stationarity. When the order of the autoregression is more than one, say x_t = b1 * x_{t-1} + b2 * x_{t-2} + epsilon_t, we need to test for roots in the autoregression’s characteristic equation. We’re not responsible for this in L2.

To take it simple, random walk and unit root is the same thing in principle. Unit root means b1 coefficient being unit or 1. And any AR(1) having a unit root (i.e. b1 = 1) is a random walk. Both are not desired in a time series regression. If there is unit root problem, solution is to first difference the series.