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time series

Innocent Iguana answered on March 9, 2023 Popularity 9/10 Helpfulness 3/10

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- Correlation = root of r-squared
- Correlation between time series:
    - if 2 different stocks are trending, their correlation is high even if they do not show same pattern
    - Correct way : find correlation between the stock returns instead (eg: correlation between daily percentage change of two stocks)
- Predicting future points using regression : dependent time series = independent time series * slope + intercept + error
- Auto-correlation:
    - correlation between a time series with a lagged version of itself
    - an "echo" that exists in all points in a time series with other points in the past
    - eg: 1,2,3,4,5,6,7 in this series second number = first number + 1, third number = second number + 1.. this exist for all points
    - Negative autocorrelation = mean reverting
        - stocks have historically negative autocorrelation over weeks
        - strategy to make money : buy down -> sell up
    - Positive autocorrrelation = momentum
        - commodities and currencies have historically positive autocorrelation over months
        - strategy to make money : buy up -> sell down
    - An autocorrelation graph 
        - shows how many past points (lags) can we use to predict the future (including the present point).
        - Shows suitable model for prediction
- White Noise
    - constant mean,variance over time, 0 autocorrelation at all lags
    - Gaussian White Noise : the white noise has gaussian distribution and show bell curve
- Random Walk and White noise
    - Stock market follow a random walk, and so the return (gain or percent change) is white noise (Yesterday price - Today price = noise)
    - You cannot forecast a random walk. The best guess : todays price is same as yesterdays price
    - random walk with drift = random walk + mean (drift)
    - How do we make sure if a series is rendom walk?
        - Dickey Fuller Test : You can test if a series is random walk
        - Augmented Dickey Fuller Test : Test if a series is random walk with more than one lags through augmentation
- Stationarity
    - mean, variance and autocorrelation of data are time invariant
    - Why we need it: non-stationary data is hard to model due to large number of parameters (new parameters found for each point in time)
    - eg: stock price (random walk) is non-stationary. reason : price of today will differ from price of 10 years into the future
    - eg: white noise is stationary. reason : mean, variance and auto-correlation of 100 data is same as 1000 data points
    - non-stationary to stationary : may require several transformations like:log transformation, take the difference between current and a lagged version of itself (the right lag = look at acf graph)
    - Regression model:
        1. AR model : 
            - Theory : The next value should retain some information from the previous VALUE
            - todays value = mean + co-efficient * yesterday's value + noise(y = mx + c)
            - co-efficient = phi. Negative phi = mean reversion, positive phi = momentum
            - -1 < phi < +1 for stationary series
            - phi = 1 for random walk (high autocorrelation) , phi = 0 for white noise (no autocorrelation) , 
            - autocorrelation decays exponentially at a rate of phi
        1. MA model : 
            - Theory: The next value should retain some information from the previous NOISE
            - todays value = mean + co-efficient * yesterday's noise + today's noise (y = mx + c)
            - co-efficient = theta. Negative theta = mean reversion, positive theta = momentum
            - stationary for all values of co-efficient, theta
            - theta = 1 for random walk (high autocorrelation) , theta = 0 for white noise (no autocorrelation) , 
            - autocorrelation decays exponentially at a rate of phi
- Partial auto-correlation : 
    - incremental benefit of adding another lag
    - quantifies how significance adding n-th lag is when there is already (n-1)th lag
- Information Criteria : adjusts penalties on number of parameters in the model. The best model has least AIC or/and BIC model among the peers.
- Cointegration model: 
    - two series can be random walk, however the distance between them may be mean reverting. 
    - Check P-cQ with dicky fuller to see if that is random walk
    - eg: Owner and his dog with a leash. steps of owner or dog is separately random walk. but the distance between them is mean reverting
        - If dog falls too far behind, it gets pulled forward
        - If dog gets too far ahead, it gets pulled back
    - eg : Natural gas vs heating oil stock movement. or, economic substitutes or competing company stocks.
- Modeling flow:
    1. See if the series is stationary (dicky fuller test, higher p-value means non-stationary or random walk)
    2. take difference / percent change (to make it stationary)
    3. compute ACF and PACF
    4. Fit few AR, MA and ARMA models
    5. compare AIC, BIC to find the best model
    6. Forecast

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Contributed on Jan 31 2024

Innocent Iguana

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It means you are able to go back in time and retrieve past values of a resource, even if the current value is not the same as it was

previously or if it has changed multiple different times.

Some examples of how you might use the time series in a Smart Contract hook:

Retrieving the latest denomination, used to check that a posting currency is valid.

Retrieving a past balance according to a provided timestamp.

Retrieving a past balance according to the previous day’send of dayy balance for that account.

Retrieving all of the flag values for a repayment holiday flag over the previous month.

Retrieving only the latest value for a repayment holiday flag.

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Contributed on Mar 09 2023

Vinay Rawal

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