# 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

$$r_t = \log \frac{p_t}{p_{t-1}}$$

The price series is obviously very autocorrelated, since today's price is yesterday's price plus small delta.

• 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.... Commented Nov 9, 2018 at 23:04
• I was trying to follow this advice and was immediately hit with NaN's due to the inability to log negative values... So I'm assuming one has to convert to absolute returns, log, then reapply the sign Commented Apr 4, 2020 at 21:36

ACF plot suggests there is autocorrelation which lasts for long time. The series is clearly not stationary. You may try differencing once - return time series, then plot boathouse ACF and PACF.

You should definitely look at Marcos Lopez De Prado, Advances in Financial Machine Learning (2018). In chapter 5 he lays out an innovative concept: Fractional Differentiation. This idea is to provide a continuum in which differencing time series is given to make time series stationary. Most formal tests for stationarity (i.e. Augmented Dickey-Fuller and other tests) are weak. The simple thing to do is to divide the time series (chronologically) in two: compute: [1] Mean and variance of the first period [2] Mean and variance of the second period Both periods should be approximately the same for both metrics to conclude one has stationarity. If you don't have an economic (theoretical) rationale for trend stationarity, then you probably should assume it is not.