Intro to Business Statistics: Time Series Analysis Forecasting with Trend and Seasonality

What This Is

Forecasting with trend and seasonality is a statistical method used to predict future values of a time series data, accounting for both long-term trends and regular fluctuations. A retail chain wants to know if average daily sales will exceed $10,000 during the upcoming holiday season, considering the trend of increasing sales over the past few years and the seasonal fluctuations due to holidays and promotions.

Key Formulas & Symbols

• Trend Equation: y = β₀ + β₁x where y = predicted value, β₀ = intercept, β₁ = slope, x = time period.
• β₀: intercept or constant term
• β₁: slope or rate of change
• x: time period (e.g., month, quarter, year)
• Seasonal Index: SI = (y - ȳ) / ȳ where y = observed value, ȳ = mean value.
• SI: seasonal index
• y: observed value
• ȳ: mean value
• Trend and Seasonal Equation: y = β₀ + β₁x + γ₁sin(2πx/12) + γ₂cos(2πx/12) where y = predicted value, β₀ = intercept, β₁ = slope, γ₁ = seasonal amplitude, γ₂ = seasonal phase, x = time period.
• γ₁: seasonal amplitude
• γ₂: seasonal phase
• Least Squares Estimation: β̂ = (X^T X)^-1 X^T y where β̂ = estimated coefficients, X = design matrix, y = response variable.
• β̂: estimated coefficients
• X: design matrix
• y: response variable
• Residual: e = y - ŷ where e = residual, y = observed value, ŷ = predicted value.
• e: residual
• y: observed value
• ŷ: predicted value
• Mean Absolute Percentage Error (MAPE): MAPE = (1/n) ∑|y - ŷ| / y where MAPE = mean absolute percentage error, n = number of observations, y = observed value, ŷ = predicted value.
• MAPE: mean absolute percentage error
• n: number of observations
• y: observed value
• ŷ: predicted value

Step-by-Step Procedure

1. State hypotheses: Formulate null and alternative hypotheses about the trend and seasonality of the time series data.
2. Choose test: Select an appropriate test statistic and critical value based on the type of trend and seasonality (e.g., linear trend, seasonal index).
3. Compute test statistic: Calculate the test statistic using the observed data and the chosen test.
4. Find p-value or critical value: Determine the p-value or critical value associated with the test statistic.
5. Compare to α: Compare the p-value or critical value to the significance level α (default = 0.05).
6. Conclude: Make a decision about the trend and seasonality based on the comparison.

Common Mistakes

• Mistake: Failing to account for seasonality in the trend equation.
• Correction: Include seasonal terms in the trend equation to capture regular fluctuations.
• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.
• Mistake: Using a linear trend equation when the data exhibits non-linear behavior.
• Correction: Use a non-linear trend equation (e.g., quadratic, cubic) to capture the underlying pattern.

Quick Practice Problems

1. A company wants to forecast sales for the next quarter using a trend equation. The observed sales data for the past four quarters are: 100, 120, 140, 160. What is the slope of the trend equation?
2. Answer: 20, The slope is calculated as the change in sales divided by the change in time (quarter).
3. A retail chain wants to evaluate the effectiveness of a marketing campaign using a seasonal index. The observed sales data for the past year are: 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320. What is the seasonal index for the first quarter?
4. Answer: 1.25, The seasonal index is calculated as the observed value minus the mean value divided by the mean value.
5. A manufacturer wants to predict the demand for a product using a trend and seasonal equation. The observed demand data for the past year are: 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320. What is the predicted demand for the next quarter?
6. Answer: 340, The predicted demand is calculated using the trend and seasonal equation.

Last-Minute Cram Sheet

1. Trend Equation: y = β₀ + β₁x
2. Seasonal Index: SI = (y - ȳ) / ȳ
3. Trend and Seasonal Equation: y = β₀ + β₁x + γ₁sin(2πx/12) + γ₂cos(2πx/12)
4. Least Squares Estimation: β̂ = (X^T X)^-1 X^T y
5. Residual: e = y - ŷ
6. Mean Absolute Percentage Error (MAPE): MAPE = (1/n) ∑|y - ŷ| / y
7. p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true
8. Use a non-linear trend equation when the data exhibits non-linear behavior
9. Include seasonal terms in the trend equation to capture regular fluctuations
10. Default significance level α = 0.05