# Causal Time Series relations

Objectives of time series analysis:
– Auto-correlation: predict based on t.
– Causal 1: Analyze impact of an event on a ts variable.
– Causal 2:Analyze impact of a ts variable on another ts variable.

Causal Time Series patterns: Challenges:
1. Explanatory and response variables may have autocorrelation within them (y & y-1, x & x-1).
2. Explanatory variables(leading indicators) may have a shift of N periods of impact on response variable. E. G. Unemployment rate has impact on crime rate but after a shift of couple months.
3. Correlation does not mean causation.

t’th-order auto-correlation: correlation of y with Y.lag(t). First order correlation is the correlation between Y(t) and Y(t-1).

Overlapping cycles: Cycles of different periods that overlapping. Example: periods having day-of-week, day-of-month and month-of-year seasonalities. Solution: Use coded variable – a new seasonality dimension. Also one can apply multiple seasonality indexes to the forecasted period. So if you want to forecast for Monday, 1st of January. you can apply decomposition, then after extracting the seasonality indices of all seasons of different overalpping periods, multiply the level-trend by the day-of-week season-index, then multiply the result by day-of-month season-index, then by month-of-year season-index.

Interaction terms: Time*LagY. Some times time itself influences the correlation between Y (response variable) and X (the explanatory variable). It could happen that Y and X may regress together differently in different time periods. So we can include the “time” factor in the (auto)regression variable. So instead of regressing Y against LagX (or X.Lag), we regress Y against LagX * Time.

Cross-correlation: between 2+ variables the change over time.
Event causal effect: check the change in acf/pacf before/after event.