Moving Averages

What You'll Learn

• Simple moving average
• Weighted moving average
• Exponential smoothing
• Choosing the right method
• Forecasting applications

Simple Moving Average (SMA)

Definition: Average of k most recent observations

Formula: MA_t = (Y_t + Y_{t-1} + ... + Y_{t-k+1}) / k

Example (3-period MA): Data: 100, 110, 105, 115, 120 MA_4 = (110+105+115)/3 = 110 MA_5 = (105+115+120)/3 = 113.3

Purpose:

• Smooth out noise
• Identify trends
• Simple forecasting

Choosing k (Period Length)

Smaller k (e.g., k=3):

• More responsive to changes
• Less smoothing
• More volatile

Larger k (e.g., k=12):

• More smoothing
• Less responsive
• Lags behind actual changes

Guidelines:

k = 3 to 5: Short-term smoothing

k = 12: Monthly data, remove seasonality

k = 7: Weekly data, smooth daily variation

Rule: Match k to seasonal period for deseasonalizing

Centered Moving Average

Problem with SMA: Places average at end of period → Lags behind data

Solution: Center the average

Example (3-period centered): MA_2 = (Y_1 + Y_2 + Y_3) / 3 (Centered on period 2)

For even periods (k=4): Use 2×4 MA to center

Best for:

• Decomposition
• Identifying trend
• Historical analysis

Not for forecasting! (Need future values)

Forecasting with SMA

Simple forecast: Next period = current MA

Example: 3-month MA = 110 Forecast for next month = 110

Limitations:

• Only predicts flat line
• Can't capture trends
• Can't capture seasonality
• Lags behind changes

Works best when:

• Data is relatively flat
• No strong trend
• Short-term forecast only

Weighted Moving Average (WMA)

Idea: Give more weight to recent observations

Formula: WMA_t = w_1×Y_t + w_2×Y_{t-1} + w_3×Y_{t-2}

Where: w_1 + w_2 + w_3 = 1

Example (3-period): Weights: 0.5, 0.3, 0.2 Data: 100, 110, 105

WMA = 0.5(105) + 0.3(110) + 0.2(100) = 52.5 + 33 + 20 = 105.5

Advantage: More responsive than SMA

Choosing weights:

• More recent = higher weight
• Weights sum to 1
• Often linear (0.5, 0.3, 0.2) or exponential

Exponential Smoothing

Most popular method!

Formula: F_{t+1} = α×Y_t + (1-α)×F_t

Where:

• F_t = forecast for period t
• Y_t = actual value in period t
• α = smoothing constant (0 < α < 1)

Interpretation: New forecast = α(latest value) + (1-α)(previous forecast)

Example: α = 0.3 Previous forecast = 100 Actual = 110

New forecast = 0.3(110) + 0.7(100) = 33 + 70 = 103

Choosing Alpha (α)

α close to 0 (e.g., 0.1):

• Heavy smoothing
• Slow to respond
• Stable forecasts
• Good for noisy data

α close to 1 (e.g., 0.9):

• Light smoothing
• Quick to respond
• Volatile forecasts
• Good for rapidly changing data

Typical values: 0.1 to 0.3 for most business data

Optimization: Choose α that minimizes forecast error (MSE)

Exponential Smoothing Example

Data: 100, 105, 110, 115 α = 0.3 Initial forecast = 100

Period 1: F_1 = 100 (initial) Actual = 100 Error = 0

Period 2: F_2 = 0.3(100) + 0.7(100) = 100 Actual = 105 Error = 5

Period 3: F_3 = 0.3(105) + 0.7(100) = 101.5 Actual = 110 Error = 8.5

Period 4: F_4 = 0.3(110) + 0.7(101.5) = 104.05 Actual = 115 Error = 10.95

Period 5 forecast: F_5 = 0.3(115) + 0.7(104.05) = 107.34

Why "Exponential"?

Weights decay exponentially:

Most recent: α One back: α(1-α) Two back: α(1-α)² Three back: α(1-α)³

Example (α=0.3):

• Recent: 0.30
• One back: 0.21
• Two back: 0.15
• Three back: 0.10

All observations have some weight! (Unlike SMA which drops old data)

Double Exponential Smoothing

For data with trend:

Two equations:

Level: L_t = α×Y_t + (1-α)×(L_{t-1} + T_{t-1})

Trend: T_t = β×(L_t - L_{t-1}) + (1-β)×T_{t-1}

Forecast: F_{t+h} = L_t + h×T_t

Parameters:

• α: level smoothing
• β: trend smoothing

Use when: Data has linear trend

Triple Exponential Smoothing (Holt-Winters)

For trend AND seasonality:

Three components:

• Level
• Trend
• Seasonal

Most complete exponential smoothing!

Use when:

• Data has trend
• Data has seasonality
• Need automatic forecasting

Popular in: Business forecasting packages

Comparing Methods

SMA: ✓ Simple ✓ Easy to understand ✗ Equal weights ✗ Lags behind

WMA: ✓ More responsive ✓ Custom weights ✗ Manual weight selection ✗ Still drops old data

Exponential Smoothing: ✓ Most responsive ✓ All data used ✓ Self-correcting ✓ One parameter ✗ Slightly harder to explain

Winner for most cases: Exponential smoothing!

Excel Implementation

SMA: =AVERAGE(B2:B4) for 3-period MA

Moving average tool: Data → Data Analysis → Moving Average

Exponential Smoothing: Data → Data Analysis → Exponential Smoothing Enter α (damping factor = 1-α)

Manual exponential: =alpha*Actual + (1-alpha)*PreviousForecast

Python Implementation

import pandas as pd
import numpy as np

# Data
data = [100, 105, 110, 115, 120, 125]
df = pd.DataFrame({'sales': data})

# Simple Moving Average
df['SMA_3'] = df['sales'].rolling(window=3).mean()

# Weighted Moving Average
weights = np.array([0.5, 0.3, 0.2])
df['WMA_3'] = df['sales'].rolling(window=3).apply(
    lambda x: np.sum(weights * x), raw=False
)

# Exponential Smoothing
alpha = 0.3
df['EXP'] = df['sales'].ewm(alpha=alpha, adjust=False).mean()

# Using statsmodels for forecasting
from statsmodels.tsa.holtwinters import SimpleExpSmoothing

model = SimpleExpSmoothing(df['sales'])
fit = model.fit(smoothing_level=0.3)
forecast = fit.forecast(steps=3)
print(f"Next 3 periods: {forecast}")

Forecast Accuracy

Measuring error:

MAE (Mean Absolute Error): Average of |Actual - Forecast|

MSE (Mean Squared Error): Average of (Actual - Forecast)²

MAPE (Mean Absolute Percentage Error): Average of |Actual - Forecast| / Actual × 100%

Use:

• Compare different α values
• Choose best forecasting method
• Optimize parameters

When to Use Each Method

Simple Moving Average:

• Quick smoothing
• No forecasting needed
• Educational purposes

Weighted Moving Average:

• Recent data more important
• Known weight structure

Exponential Smoothing:

• Most common choice!
• Automatic forecasting
• Adapts to data
• Works for various patterns

Double/Triple Exponential:

• Trend present
• Seasonality present
• Automated forecasting system

Limitations

All moving averages:

• Only use past data
• Can't predict turning points
• Assume patterns continue
• No confidence intervals

Not good for:

• Long-term forecasts
• Structural changes
• External factors
• Causal modeling

Practical Example

Monthly sales forecasting:

Data (6 months): Jan: 100, Feb: 110, Mar: 105, Apr: 115, May: 120, Jun: 125

Method 1: 3-month SMA Forecast for Jul = (115+120+125)/3 = 120

Method 2: Exponential (α=0.3) Build up from Jan: ... Forecast for Jul = 118

Method 3: Naive (last value) Forecast for Jul = 125

Compare accuracy: Calculate MAE for past months Choose method with lowest error

Practice Exercise

Sales data: Week 1: 50 Week 2: 55 Week 3: 60 Week 4: 58 Week 5: 62

Tasks:

1. Calculate 3-week SMA for Week 5
2. Forecast Week 6 using SMA
3. Calculate exponential smoothing forecast for Week 6 (α=0.2, initial=50)
4. Which gives higher forecast?

Answers:

1. SMA_5 = (60+58+62)/3 = 60
2. Week 6 forecast = 60
3. Build up: 50, 51, 52.8, 54.64, 56.11, 57.29 Week 6 forecast = 57.29
4. SMA gives higher forecast (60 vs 57.29)

Next Steps

Learn about Naive Methods (forecasting)!

Tip: Start simple with SMA, then try exponential smoothing for better results!