1. AAMP Training Materials
Module 3.1: Analyzing Price Seasonality
Asfaw Negassa (CIMMYT) & Shahidur Rashid (IFPRI)
This is the first section of the AAMP Price Analysis module.
Exercises for this module are found in the following Excel file:
AAMP Module 3.1 – Analyzing Price Seasonality.xls

2. Objectives
Understand the role of seasonality analysis in economic time series data
Review basic concepts of seasonality analysis
Compute seasonal indices
Forecast future prices using seasonal indices

3. Background
Why Seasonality Analysis?
Basic concepts

4. Why undertake seasonality analyses?
Economic time series variables (e.g., prices, sales, purchase, etc.) are composed of various components
One of the critical components is seasonality, especially in cases of:
Agriculture production
Marketing of goods
Commodity pricing

5. Why undertake seasonality analyses?
Understanding patterns of movement in time series variables is useful for making better forecasts in
Marketing decisions
When to buy?
When to sell?
How long to store?
Production decisions
Seasonality in rainfall (timing of different operations)
Seasonality in insect infestation (timing of insecticide application)
Food security interventions
Determine periods when households are most vulnerable (e.g., seasonal price highs)
Monitoring changes in households food security situations (e.g., deviations from known seasonal price patterns)

6. Basic concepts
A time series variable (e.g., prices, sales, purchases, stocks, etc) is composed of four key components:
Long-term trends (T)
Seasonal components (S)
Cyclical components (C)
Irregular or random components (I)
These components, when examined individually can help to better understand the sources of variability and patterns of time series variables – hence time series decomposition
The four components in detail:
Long-term trends: Over time, prices of goods can acquire a general long-term trend, either up or down, which should be accounted for when analyzing price seasonality. For example, as a country begins to adopt new technology in the planting, harvesting and storage of a crop, prices decline over the long term.
Seasonal components: Particularly common with prices for commodities such as maize, prices tend to take on a notable seasonal pattern over a one year period, rising as the stocks in the country are depleted, then dropping quickly as the harvest comes in.
Cyclical components: This is wavelike upward and downward movements of the time series data around the long-term trend. Unlike the seasonal component, this component is of longer duration and less regular. These wavelike movements are caused by ups and downs in general level of business activities in the economy.
Irregular components: These are random fluctuations in series data which are hard to predict. This is the residual component.

7. Basic concepts Pt = Tt x St x Ct x It
There are several ways of decomposing time series variables (e.g., additive model, multiplicative model)
The basic multiplicative model is given as:
Pt = Tt x St x Ct x It
Where:
Pt is the time series variable of interest
Tt is the long-term trend in the data
St is a seasonal adjustment factor
Ct is the cyclical adjustment factor
It represents the irregular or random variations in the series
People commonly use the multiplicative model because it is relatively easy to decompose time series variables into its various components.

8. Time series price variables
Include long term, cyclical, seasonal and irregular trends

9. Computing seasonal indices
There are several different techniques which are used to isolate and examine individually the different components of time series variables
Here we focus on the ratio-to-moving average method, which is commonly used

10. Computing seasonal indices
Step 1: Deseasonalizing
Remove the short-term fluctuations from the data so that the long-term and cyclical components can be clearly identified
The short-term fluctuations include both seasonal (St) patterns and irregular (It) components
The short-term fluctuations can be removed by calculating an appropriate moving average (MA) for the series
Assuming a 12-month period, the moving average for a time period t (MAt) is calculated as:
MAt = (Pt – 6 + … + Pt + … + Pt + 5) / 12

11. Computing seasonal indices
Step 1: Continued….
For monthly data, the number of periods (12) is even and is not centered – need to center it
To center the moving averages, a two-period moving average is calculated as follows:
CMAt = (CMAt + CMAt +1) / 2………
= Tt x Ct……...
Seasonal and irregular components are removed

12. Centered Moving Average (CMA)
Seasonal and irregular components removed
The figure above shows the deseasonalized price for maize in Dar es Salaam, Tanzania over time. The data is monthly, so the CMA is calculated by centering a moving average of 12 months. The CMA removes short-term fluctuations in the data caused by seasonal and irregular factors. What remains is long-term trends and cyclical factors.

13. Computing seasonal indices
Step 2: Measuring the degree of seasonality
The degree of seasonality is measured by finding the ratio of the actual value to the deseasonalized value
SFt = Pt / CMAt
= Tt x St x Ct x It / Tt x Ct
= St x It
Where SFt is the seasonal factor and others are defined as before.

14. Actual price (P) / Deseasonalized price (CMA)
Seasonal Factor (SF)
Actual price (P) / Deseasonalized price (CMA)
The Seasonal Factor (SF) describes short-term fluctuations in price attributable to seasonality and irregular factors. All that remains is to remove the irregular factors, and a picture of seasonal price variability (S) emerges.

15. Computing seasonal indices
Step 3: Establishing average seasonal index
This is obtained by taking the average of seasonal factors for each season
Take the sum of SFs for the month of January and divide by the number of SFs for January over the entire data period
Pure seasonal index (S) obtained, irregular component removed
Note: the sum of indices for all months add-up to 12.
Issues:
Predictability of seasonal patterns
Changes in seasonal patterns
Further descriptive and regression analysis of seasonal indices can help to discern changes in seasonal patterns

16. Seasonal index (S): Multiple years
Pure seasonal component (S) without irregular component
The Seasonal Index (S) above was composed by averaging the monthly Seasonal Factors (SF). The sum of all January SFs divided by the number of January SFs in the sample give the January Seasonal Index (S). This is repeated for each month until the above pattern emerges.
The following slide shows the Seasonal Index for a single year.

17. Seasonal Index (S): One year
The seasonal Index helps predict price extremes
Seasonal High
Seasonal Low
Maize prices in Dar es Salaam are highest in March and lowest in July. This corresponds to the Masika rainy season (the long rains) which begin in March and end in June. Also , the seasonal highs and lows allow to determine the length of storage period between the seasonal low and high if storage is profitable given the storage costs over the defined storage period.

18. Computing seasonal indices

19. Completing the price decomposition
In order to complete the price decomposition, three components must be computed
Long term time trend of deseasonalized data (CMAT)
Cyclical Factor (CF)
Irregular Factor (I)
The combination of CMAT and S is a useful method for forecasting future prices

20. Computing other components
Step 4: Finding the long-term trend
The long-term trend is obtained from the deseasonalized data using OLS as follows:
CMAt = a + b (Time)
Where Time = 1 for the first period in the dataset and increases by 1 each month thereafter
Once the trend parameters are determined, they are used to generate an estimate of the trend value for CMAt for the historical and forecast period.

21. OLS on deseasonalized price (CMA)
Long term trend (CMAT)
OLS on deseasonalized price (CMA)

22. Computing other components
Step 5: Finding the cyclical component (CF)
The CF is given as the ratio of centered moving average (CMAt) to the centered moving average trend (CMATt) as follows:
CF = CMAt / CMATt
= Tt x Ct x It / Tt
=Ct

23. Deseasonalized price (CMA) / Long-term trend (CMAT)
Cyclical Factor (CF)
Deseasonalized price (CMA) / Long-term trend (CMAT)

24. Computing other components
Step 6: Finding the irregular component (It)
The It is given as the ratio of seasonal factor (SF) to pure seasonal index (S) as follows:
It = St It / St
= It
This completes the decomposition

25. Irregular component (I)
Seasonal Factor (SF) / Seasonal Index (S)

26. Forecasting using seasonality analysis
Forecasting time series variables
Prices for the next year can be forecasted
Timing of price changes
When will prices be low?
When will prices be high?
Magnitude of price level at specific future dates
Magnitude of temporal price differential (between seasonal high and low)
Is storage profitable?

27. Forecasting using seasonality analysis
Two main ways of forecasting
Forecast monthly values by multiplying estimated average value for the next year by the seasonal index for each month– this assumes no significant trend,
First estimate the 12-month trend for deseasonalized data and then apply the seasonal index to forecast the actual prices for the next year
Forecast Pt = Tt x St

28. Forecasting using method 2
CMAT x S
In the figure above, the black and red lines are exactly the same where all four components of Price are included:
Pt = Tt x St x Ct x It
The lines deviate when forecasting because the irregular component and the cyclical component are missing from the calculation:
Forecast Pt = Tt x St
Clearly, there are limitations to the accuracy of such forecasts.

29. Forecasting limitations
The seasonal analysis is used under normal conditions and there are several factors which alter the seasonal patterns
Drought, floods, earthquake, etc.
Government policy changes
Abnormal years should not be included in the computation of seasonal indices

30. Exercises Open Module 3.1 Excel workbook
Read over the notes in the [NOTES] sheet
In [Dar Maize Price Analysis] sheet, replicate the outcomes in the [Tanzania Example]
First, calculate seasonality indices (MA, CMA, SF and S)
Then, forecast future prices (CMAT, CF, predicted price)
Make sure to understand what each component does
Don’t just copy and paste the formulas.

31. References
Tschirley, David Using microcomputer spreadsheets for spatial and temporal price analysis: an application to rice and maize in Ecuador. In Prices, Products, and people: Analyzing agricultural markets in developing countries (Scott, G., editor) International Potato Centre, Lima, Peru.