Chapter 7

Forecasting and Time Series

📖 ~55 min read 📈 2 interactive charts ✍️ 2 practice questions 🎯 1 linked game

7.1 Time Series Components

A time series is a sequence of data points collected at regular time intervals — monthly revenue, daily stock prices, weekly website traffic. Forecasting involves using past patterns in this data to predict future values. Before building any forecasting model, you must understand the four fundamental components that shape time series behavior.

The Four Components

  1. Trend: The long-term upward or downward movement in the data. A company experiencing steady growth over five years has an upward trend.
  2. Seasonality: Regular, predictable patterns that repeat within a fixed period (typically within a year). Retail sales spike every holiday season; ice cream sales peak in summer.
  3. Cyclical: Longer-term fluctuations that do not have a fixed period. Business cycles of expansion and contraction typically span multiple years and are harder to predict than seasonal patterns.
  4. Irregular (Random): Unpredictable, short-term fluctuations caused by random events — a supply disruption, a viral social media post, or an unexpected competitor move.

Additive vs. Multiplicative Models

Time series components can be combined in two ways:

  • Additive: Y = Trend + Seasonal + Cyclical + Irregular. Use when seasonal swings are roughly constant in magnitude regardless of the level of the series.
  • Multiplicative: Y = Trend × Seasonal × Cyclical × Irregular. Use when seasonal swings grow proportionally with the level of the series.
NorthStar Enterprises — Quarterly Revenue (5 Years)
🏪 NorthStar Enterprises

The chart above shows NorthStar's quarterly revenue over five years. Three components are visible: a clear upward trend as the company grows, a seasonal pattern with Q4 consistently the highest quarter (holiday demand) and Q1 the lowest, and some irregular variation from quarter to quarter. The seasonal peaks also grow larger over time, suggesting a multiplicative model may be more appropriate than an additive one.

✓ Check Your Understanding
Which time series component repeats within a fixed period, typically within a year?
Trend
Cyclical
Seasonal
Irregular

7.2 Moving Averages and Exponential Smoothing

The simplest forecasting methods smooth out short-term noise to reveal the underlying pattern. Two foundational techniques are the simple moving average (SMA) and exponential smoothing (ES).

Simple Moving Average

A moving average forecasts the next period by averaging the most recent n observations. A 3-month moving average, for example, forecasts next month's value as the average of the last three months. Larger values of n produce smoother forecasts but respond more slowly to changes.

Simple Moving Average (SMA)
📊 Excel: =AVERAGE(last_n_values)
where n is the number of periods in the moving average window and yt is the observed value at time t.

Exponential Smoothing

Exponential smoothing addresses a key limitation of the moving average: it treats all past observations in the window equally. In many business settings, recent data should matter more than older data. Exponential smoothing assigns exponentially decreasing weights to older observations, controlled by a smoothing constant (alpha) between 0 and 1.

Exponential Smoothing
📊 Excel: =alpha*y_t + (1-alpha)*F_t
where Ft+1 is the forecast for the next period, α is the smoothing constant (0 < α < 1), yt is the actual value at time t, and Ft is the forecast for time t.
Mean Absolute Error (MAE)
📊 Excel: =AVERAGE(ABS(actual-forecast))
MAE measures the average absolute forecast error. Lower MAE indicates a more accurate forecasting method.
✎ Worked Example: 3-Month MA vs. Exponential Smoothing (α = 0.3)
1
NorthStar's last 6 months of revenue ($K):
Jan: 180, Feb: 195, Mar: 210, Apr: 190, May: 220, Jun: 215
2
3-month MA forecast for July:
FJul = (190 + 220 + 215) / 3 = 208.3
3
Exponential smoothing forecast for July (using FJun = 205.8):
FJul = 0.3 × 215 + 0.7 × 205.8 = 64.5 + 144.1 = 208.6
4
Result: Both methods forecast about $208–209K for July. To choose between them, compare their MAE over a holdout period.
Actual vs. Moving Average vs. Exponential Smoothing
🎮
Forecasting League Compete to produce the most accurate forecasts using MA and ES methods
✓ Check Your Understanding
In exponential smoothing, a higher value of α (smoothing constant) means:
More weight on older observations
More weight on recent observations
Larger forecast errors
Smoother (less responsive) forecasts
💡 Key Takeaway

Higher α makes forecasts more responsive to recent changes but also more volatile. Lower α produces smoother forecasts that are slower to adapt. The right choice depends on how quickly the underlying pattern changes in your business context.

7.3 Trend and Regression Forecasting

When a time series has a clear trend, simple moving averages and exponential smoothing will systematically lag behind the data — always underforecasting in an upward trend and overforecasting in a downward trend. Trend-based regression addresses this by fitting a line (or curve) to the time series and projecting it forward.

Linear Trend via Regression

The simplest trend model treats time as the predictor variable: Y = b0 + b1 × t. The slope b1 represents the average per-period change in the series. In Excel, this is easily accomplished with the FORECAST or TREND function, or by adding a trendline to a chart.

Seasonal Adjustment

When both trend and seasonality are present, a two-step approach works well: (1) remove seasonality by computing seasonal indices and dividing them out, (2) fit a trend line to the deseasonalized data, and (3) multiply the trend forecast by the appropriate seasonal index to get the final prediction.

Combining Methods

In practice, analysts often compare multiple forecasting methods on a holdout sample. The method that produces the lowest MAE (or MAPE, or RMSE) on the holdout period is selected for production forecasting.

💡 Key Takeaway

No single forecasting method dominates in all situations. Always compare methods using holdout data. Simple methods like moving averages work well for stable series, while regression-based approaches are better when a strong trend is present. The best forecasters combine quantitative methods with domain knowledge about the business.

7.4 Chapter Summary

Time series forecasting is one of the most directly useful statistical skills in business. From inventory planning to sales targets to budgeting, the ability to project future values from historical patterns drives decisions across every function.

💡 Chapter 7 Summary

Components: Every time series can be decomposed into trend, seasonality, cyclical, and irregular components. Understanding these components is the first step in any forecasting exercise.

Smoothing Methods: Simple moving averages and exponential smoothing are foundational techniques. SMA treats all recent observations equally; ES gives more weight to recent data through the smoothing constant α.

Trend Forecasting: When a clear trend is present, regression-based approaches outperform simple smoothing. Seasonal adjustment allows you to handle both trend and seasonality simultaneously.

📋 Chapter 7 — Formula Reference
Measure Formula Excel Function
SMA
=AVERAGE(last n values)
Exponential Smoothing
=alpha*y_t+(1-alpha)*F_t
MAE
=AVERAGE(ABS(errors))
Linear Trend
=FORECAST(x, known_y, known_x)
Up Next
Chapter 8: Monte Carlo Simulation