# Interpreting ACF

I am currently struggling with the interpretation of a price chart and the corresponding ACF graph. The question is, if there is momentum in the price of this asset. This is the corresponding price chart for a period of 19 years (5000 business days):

It doesn't´t seem to have much of momentum when looking at the price development. After verifying that the time series is (trend)-stationary by means of the Zivot / Andrews Test (ur.za in R), i generated the ACF plot to get a further idea of potential Momentum. And there´s the problem. The ACF graph indicates a price continuation pattern of around 700-800 lags (business days as the data has business days as frequency) or 2.5 - 3 years of momentum. But this is in strong contrast to the price chart above and to the efficient market hypothesis. Is there any rationale mistake from my side?

You need to compute the autocorrelation of the log returns $$r_t$$, not of the prices, $$p_t$$. The relationship of the log return series to the price series is
$$r_t = \log \frac{p_t}{p_{t-1}}$$
• To be more precise, log return $r_t = \log \left( \frac{P_t + D_t}{P_{t-1}} \right)$ where $D_t$ denotes dividends (and/or any other distributions). A problem working directly in prices is that it probably contains a unit root, hence $\{P_t\}$ isn't stationary, hence, a time-invariant mean $\mu = E[P_t]$ doesn't exist, hence a time-invariant autocorrelation function doesn't exist either etc.... – Matthew Gunn Nov 9 '18 at 23:04