Practical Time Series Analysis

Zero mean models

The zero-mean models have a constant mean and constant variance and shows no predictable trends or seasonality. Observations from a zero mean model are assumed to be independent and identically distributed (iid) and represent the random noise around a fixed mean, which has been deducted from the time series as a constant term.

Let us consider that X₁, X₂, ... ,X_n represent the random variables corresponding to n observations of a zero mean model. If x₁, x₂, ... ,x_n are n observations from the zero mean time series, then the joint distribution of the observations is given as a product of probability mass function for every time index as follows:

P(X1 = x1,X2 = x2 , ... , Xn = xn) = f(X1 = x1) f(X2 = x2) ... f(Xn = xn)

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