Intro to Business Statistics: Time Series Analysis - Components of Time, Series Trend Seasonal Cyclical Irregular

What This Is

Time series analysis is crucial in business decisions as it helps identify patterns and trends in data over time. A retail chain wants to know if average daily sales exceed $10,000 during the holiday season. By understanding the components of time series, they can make informed decisions about inventory, staffing, and marketing strategies.

Key Formulas & Symbols

• Trend: The long-term pattern in a time series, often represented by a linear or non-linear equation.
• Trend Equation: y = + x + ?, where y = dependent variable, x = independent variable, = intercept, = slope, and-= error term.
• Seasonal Component: The regular fluctuations in a time series that occur at fixed intervals, such as monthly or quarterly.
• Seasonal Index: A measure of the relative magnitude of the seasonal component, calculated as the average of the seasonal component divided by the average of the time series.
• Cyclical Component: The long-term fluctuations in a time series that occur over several years or decades.
• Cyclical Index: A measure of the relative magnitude of the cyclical component, calculated as the average of the cyclical component divided by the average of the time series.
• Irregular Component: The random fluctuations in a time series that are not predictable.
• Irregular Index: A measure of the relative magnitude of the irregular component, calculated as the average of the irregular component divided by the average of the time series.
• Autocorrelation Function (ACF): A measure of the correlation between a time series and its lagged values.
• ACF Formula: ?(k) = Cov(Xt, Xt-k) / (?X^2), where ?(k) = autocorrelation at lag k, Cov(Xt, Xt-k) = covariance between Xt and Xt-k, and ?X^2 = variance of the time series.
• Partial Autocorrelation Function (PACF): A measure of the correlation between a time series and its lagged values, while controlling for the effects of intermediate lags.
• PACF Formula: ?(k) = Cov(Xt, Xt-k | Xt-1, Xt-2, ..., Xt-k+1), where ?(k) = partial autocorrelation at lag k.

Step-by-Step Procedure

1. Identify the Time Series: Determine the type of time series (e.g., sales, temperature, stock prices) and the frequency of the data (e.g., daily, monthly, quarterly).
2. Plot the Time Series: Visualize the time series to identify patterns, trends, and seasonality.
3. Calculate the ACF and PACF: Compute the ACF and PACF to identify the order of the autoregressive (AR) and moving average (MA) components.
4. Determine the Model: Based on the ACF and PACF, determine the order of the AR and MA components and select a suitable time series model (e.g., ARIMA, SARIMA).
5. Estimate the Model Parameters: Use the selected model to estimate the parameters (e.g., coefficients, variances).
6. Evaluate the Model: Assess the goodness of fit, residual plots, and other diagnostic checks to ensure the model is adequate.

Common Mistakes

• Mistake: Failing to account for seasonality and cyclical components in the time series.
• Correction: Use techniques such as deseasonalization and detrending to remove these components and improve model accuracy.
• Mistake: Ignoring the autocorrelation structure of the time series.
• Correction: Use the ACF and PACF to identify the order of the AR and MA components and select a suitable time series model.
• Mistake: Failing to evaluate the model's goodness of fit and residual plots.
• Correction: Use diagnostic checks such as the Ljung-Box test and residual plots to ensure the model is adequate.

Quick Practice Problems

1. A company wants to forecast quarterly sales using an ARIMA model. The ACF and PACF indicate an AR(1) component. What is the order of the AR component?
2. Answer: 1, as indicated by the ACF and PACF.
3. Explanation: The ACF and PACF indicate a single lagged term, suggesting an AR(1) component.
4. A retailer wants to analyze the sales of a new product over time. The time series exhibits a strong seasonal component. What type of index should be used to measure the seasonal component?
5. Answer: Seasonal index.
6. Explanation: A seasonal index measures the relative magnitude of the seasonal component, which is essential in this scenario.
7. A financial analyst wants to forecast stock prices using a SARIMA model. The ACF and PACF indicate a SARIMA(1,1,1)(1,1,1)12 component. What is the order of the seasonal AR component?
8. Answer: 1.
9. Explanation: The ACF and PACF indicate a single lagged term in the seasonal AR component.

Last-Minute Cram Sheet

1. p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true.
2. ACF and PACF are used to identify the order of the AR and MA components.
3. Seasonal index measures the relative magnitude of the seasonal component.
4. Cyclical index measures the relative magnitude of the cyclical component.
5. Irregular index measures the relative magnitude of the irregular component.
6. ARIMA models are used for non-seasonal time series, while SARIMA models are used for seasonal time series.
7. Autocorrelation structure is essential in time series modeling.
8. Goodness of fit and residual plots are used to evaluate the model's adequacy.
9. Ljung-Box test is used to check for autocorrelation in the residuals.
10. Failing to account for seasonality and cyclical components can lead to inaccurate forecasts.