Univariate time series analysis Flashcards
Univariate Time Series Analysis
Univariate time series analysis is a statistical method used to study the behavior of a single variable observed sequentially over time. Unlike cross-sectional data, which captures information at a specific point, time series data consists of observations taken at regular intervals (e.g., daily, monthly, quarterly). The primary objective of univariate time series analysis is to identify patterns such as trends, seasonality, and autocorrelation to make forecasts or understand past behavior.
Key Features of a Time Series
1.Trend: A long-term increase or decrease in the data. For example, GDP tends to grow over time.
2.Seasonality: Regular fluctuations occurring at fixed intervals due to seasonal factors. For example, retail sales often peak in December.
3.Cyclic Patterns: Long-term oscillations not tied to fixed intervals, usually influenced by business cycles.
Autoregressive (AR) Models
An autoregressive model expresses the current value of the series as a linear combination of its past values. This means that past values influence the present.
Yt = δ1Yt-1 + δ2Yt-2 + …. δpYt-p + ξt
Where:
Yt is the current value of the time series,
𝛿𝑖 are the autoregressive coefficients,
p is the order of the AR process, indicating how many past values are used,
𝜉𝑡 is a white noise error term.
Moving Average (MA) Models
A moving average model represents a time series as a function of past error terms (shocks). It captures short-term dependencies in the data. The general MA(q) process is given by:
Yt = ξt + θ1ξt-1 + θ1ξt-2 + …. θqξt-q
Where:
ξt is a white noise term,
𝜃𝑗 are the moving average coefficients,
q is the order of the MA process, indicating how many past error terms are used.
Autoregressive Moving Average (ARMA) Models
The ARMA model combines AR and MA models, capturing both the dependence on past values and past error terms. ARMA models are used when the time series is stationary. If the series is non-stationary, it needs to be differenced first.
Autoregressive Integrated Moving Average (ARIMA) Models
When a time series is non-stationary, an ARIMA model is used, which includes a differencing step to remove trends. The ARIMA (𝑝,𝑑,𝑞) model is given by:
What is the null hypothesis in a Dickey-Fuller test?
The time series has a unit root and is non-stationary.
What does it mean if the ACF decays slowly?
The time series is likely non-stationary.
What is seasonality in time series data?
Regular fluctuations that occur at fixed intervals due to seasonal factors.
What are cyclic patterns in time series data?
Long-term oscillations influenced by business cycles but not tied to fixed intervals.
