Time Series - AR(1) Introduction
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Introduction
Times series. Simply put, when data is collected repeatedly over time, it often becomes a time series problem. It is extremely commonly used in academia and in industry, as this analysis can forecast future values, explain how the past influences the future, and highlight changes over time. Ranging from manufacturing quality control to financial market predictions, time series is well represented everywhere.
Arguably one of the most basic time series models is the \(1^{st}\) order autoregression model — AR(1). The intuition behind the model is simple:
The value at time t is based on the value from the previous time, t-1, and random noise.
Typically speaking, time series data has some sort of relationship with previous values, that’s what makes it unique and powerful. By modeling values based on the past, we can make use of the properties of a time series.
Naive Approach
If you ignore the time aspect of the data and proceed onwards with a linear regression, you make the assumption that the time and order of observations has no relevance to the data. While there are situations where this may be valid, more often than not it breaks the independence condition of the linear regression. Additionally, it hardly ever feels like a good idea to remove such a key variable!
Assumptions
Assumptions are discussed on the Penn State lesson notes. This includes the idea that the data should be a weakly stationary series.
Weakly stationary series implies that the series has a constant mean (i.e. no trending upwards or downwards), the variance is constant, and the correlation between a time period t and t-1 is constant for all time points.
This means that if there are trends, seasonality, or certain cycles, the AR(1) may not be the optimal time series model.
Resources
Time Series Analysis with Applications in R by Cryer and Chan
Time Series Analysis and Its Applications With R Examples by Shumway and Stoffer