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Time Series Models

While standard regression we must assume observations are independent from one another, but with time series data we expect that neighboring observations are correlated. Time series analysis helps organizations understand the underlying causes of trends or systemic patterns over time. A time series is simply a set of statistics that is collected at regular intervals.intervals which we can use too obtain valid inferences. Ex. the daily number of live births or death.

A Single Observation of  Stochastic Process

A Stochastic process is a (possibly) infinite sequence of variables ordered in time {Y0, Y1, Y2 ...}. A time series is a single realization of a stochastic process. We want to make inference about the properties of the underlying stochastic process from a single observation.

There are two assumptions in time series analysis:

  1. The data sequence is stationary. This means if all the times are shifted by the same amount, the probability distribution remains the same; meaning it depends on relative and not absolute values. In other terms:
    $$ (X_{t_1}, ... , X_{t_k}) =(X_{t_{1 + h}}, ... , X_{t_{k + h}}) $$
    for all time points t and integer h
    Under this assumption we can use the replication over time to make inferences about the common mean, variance, and other statistics. Additionally, the degree of independence increases as the time interval between two observations increases.
  2. Ergodicity - the ability to make valid probability statements by looking over time rather than across replicates across one time.
Auto-correlation

A consequence of independence when observations are far apart enough in time is that we can use the auto-correlation function as a measure of dependence of the observations over time.

Let us define the covaraiance between time points at time k as: \(\gamma(k) = Cov(Y_{k+1}, Y_1) = Cov(Y_{k+2}, Y_2}

The sequence \( \gamma_k

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