Time Series Basics: Concepts, Forecasting Approaches, and Key Components

Time plays a major factor in many decisions across organizations.
Examples:
- Retail scheduling: retailers hire more employees in the fourth quarter due to holiday sales.
- UPS increases hires in Q4 to meet higher shipping demand.
- Healthcare experiences enrollments in the fall (around October–November).
Takeaway: time-based factors influence outcomes and decision-making; capturing this impact is valuable for planning.

Forecasting uses past experience and knowledge (your own, competitors, and other learnings) to predict the future.
Two main challenges:
1) Developing the knowledge and models is hard; tools exist but the knowledge isn’t easy to build.
2) Business changes over time (e.g., retail industry in shambles) reduce the predictive power of past behavior.
Nevertheless, time series forecasting is highly sought after because any insight (even around 20–25%) is valuable when planning for the future.
Without insight, organizations face uncertainty during time periods of change.

Judgmental forecasting:
- Based on experience and personal judgment; often lacks a formal scientific basis.
Extrapolation (quantitative):
- Uses time series data to extract the time-specific impact on the output.
- Techniques include moving averages, cumulative moving averages, seasonality extraction, and seasonality indices.
Econometrics:
- Uses regression to model how time (and other variables) affects the dependent variable.
- Can include variables like advertising, market changes, product development, etc.

Data source: past time series data (e.g., quarterly sales data).
Steps in the example workflow:
- Calculate a moving average: $MAt = \frac{1}{n} \sum{i=0}^{n-1} y_{t-i}$
- Compute a cumulative moving average: $CMAt = \frac{1}{t} \sum{i=1}^{t} y_i$
- Identify seasonality from the series: extract seasonal patterns within the time frame.
- Develop a seasonality index: $I_j = \frac{\text{Average in season } j}{\text{Overall average}}$
- Deseasonalize the data: $yt^* = \frac{yt}{I_j}$ where j indicates the season/quarter (e.g., Q1, Q2, Q3, Q4).
- Fit a regression/forecast model on the deseasonalized data to obtain a fitted value: \hat{y}_t^{reg}.
- Forecast and re-seasonalize using the seasonality index: the forecast is based on the regression fit, then scaled back by the appropriate season index.
Example of seasonality indices (quarters):
- Q1: 1.0 (baseline)
- Q2: 1.1 (10% above average)
- Q3: 1.05 (5% above average)
- Q4: 0.96 (4% below average)
- These indices show how much sales deviate from the average within a year.
Visual outcome you would see:
- Historical data (in black) and forecast (in red) on a chart.
- Example dataset described as fifteen years of data used to forecast four quarters for year 2017 and four for 2018.
Summary of the process: deseasonalize → regression/fitted value → forecast → reapply seasonality to obtain final forecast.

Perceived as simplistic because it focuses mainly on time without other drivers.
Cannot easily incorporate other factors like new product development, market changes, or other external influences.
Limitations arise because the model is primarily time-based and may miss important drivers of change.

Econometric models use regression to quantify how time and other variables affect the dependent variable (e.g., sales).
They allow including multiple explanatory variables (e.g., monthly advertising, promotions) to explain changes.
They are more flexible than simple moving-average methods but come with challenges.

Mathematical and identification challenges: it can be hard to determine what is actually causing observed changes.
Lag effects: sometimes the effect of a variable is not immediate.
- Example discussed: spending $50,000 on advertising in Quarter 1 might not produce sales increases until Quarter 2 due to lag.
Autocorrelation: current values can be correlated with past values, complicating inference and model validity.
- Inventory example: if a product is out of stock today, it is likely to be out of stock tomorrow, creating a dependency between consecutive observations.
- Autocorrelation can distort standard statistical assumptions and forecasting if not addressed.
The recording suggests that autocorrelation will be tackled in a future video; for now, focus on understanding the basic concepts.

Trend:
- How the dependent variable changes over time, across an extended period (e.g., 15 years).
- Question to answer: Is the variable increasing, decreasing, or stable over time?
Seasonality:
- A repeating pattern within a defined time period, typically within a year (e.g., quarters).
- Within-year patterns: e.g., Quarter 1 to Quarter 2, Quarter 2 to Quarter 3, Quarter 3 to Quarter 4 changes indicate seasonality.
- Could be defined over other periods (e.g., one week). Example question: do we sell more on Mondays vs. Wednesdays?
Cycle (cyclical component):
- Similar to seasonality but with a longer time frame.
- Example: economic cycles spanning a decade, such as recessions roughly every ten years.
- A cycle is a longer-term fluctuation that is not tied to a fixed calendar period.

Forecasting provides actionable insight even if imperfect (e.g., a partial forecast can guide inventory, staffing, and marketing decisions).
Understanding time series components (trend, seasonality, cycle) helps in selecting appropriate forecasting methods and interpreting results.
Recognizing limitations (lag, autocorrelation, and missing drivers) points to when econometric models or more sophisticated approaches may be warranted.
The discussion foreshadows more advanced econometric techniques that place less emphasis on purely time-based factors and more on causal drivers.

Moving average (n-period): $MAt = \frac{1}{n} \sum{i=0}^{n-1} y_{t-i}$
Cumulative moving average: $CMAt = \frac{1}{t} \sum{i=1}^{t} y_i$
Seasonality index (per season j): $Ij = \frac{\bar{y}j}{\bar{y}}$
- where \bar{y}_j is the average in season j and \bar{y} is the overall average
Deseasonalized value: $yt^* = \frac{yt}{I_j}$
Forecast after deseasonalizing and fitting: $\hat{y}{t+h}^{reg}$ is the regression forecast on the deseasonalized data, then reapply seasonality: $\hat{y}{t+h} = \hat{y}{t+h}^{reg} \cdot I{j+h}$

This set of notes captures the key ideas, methods, examples, and terminology presented in the recording.
It emphasizes the rationale for time-based forecasting, the main forecasting approaches discussed, the specifics of an extrapolation-based workflow with seasonality, and the core concepts of trend, seasonality, and cycle.