Working with seasonal data

Outline:

1. Sources of seasonality
2. Naive seasonal models
3. Seasonal adjustment
4. Seasonal decomposition

Sources of seasonal supply/demand for many products and services:

• Holidays (Christmas shopping, Thanksgiving travel & turkeys, summer tourism, greeting cards, flowers, etc.)
• Weather (e.g., demand for heating oil, gasoline, electricity, snow blowers, auto batteries, chain saws,...)
• Manufacturing year (e.g., autos)
• Fashion year (Fall/Spring clothing styles)
• Academic year ("back to school")
• Sports year (athletic gear)
• Agricultural year (farming activities, supplies of seasonal produce)

Seasonal patterns may be stable or may change over time with consumer preferences

First cut at modeling seasonality: "Naive" seasonal models

• Seasonal random walk
• Seasonal random trend

Recall:

If the first difference (month-to-month) change in Y looks like "noise," then Y is a random walk.

• Actual monthly change is random with the same distribution each month

• Predicted monthly change = average monthly change:

Y(t) - Y(t-1) =, or equivalently,

Y(t) = Y(t-1) +

Example:

Y(Sep'96) = Y(Aug'96) +

Y(Oct'96) = Y(Sep'96) +

... etc., where alpha is the average month-to-month change

Random walk (with growth) in Statgraphics:

Model type = ARIMA

Differencing:

• Nonseasonal order = 1
• Seasonal order = 0

AR, MA, SAR, SMA = 0

Constant = On

If Y is strongly seasonal....

...suppose the seasonal difference of Y looks like noise.

Then each season's data is an independent random walk--i.e., Y is a seasonal random walk.

• Actual year-to-year change is random w/ same distribution in each season of year

• Predicted year-to-year change = average year-to-year change:

Y(t) - Y(t-12) =, or equivalently

Y(t) = Y(t-12) +

....where alpha is the average year-to-year change.

Example:

Y(Sep'96) = Y(Sep'95) +

Y(Oct'96) = Y(Oct'95) +

etc.

Thus, if the average yearly increase in sales is 20,000 units, we predict that this September's sales will be 20,000 more than last September's, this October's sales will be 20,000 more than last October's, and so on. This model will perform well if growth is very steady, but it will tend to be thrown off by cyclical upturns and downturns in which one year is consistently above or consistently below what would have been predicted from the previous year.

Seasonal Random Walk in Statgraphics:

Model type = ARIMA

Differencing:

• Nonseasonal order = 0
• Seasonal order = 1

AR, MA, SAR, SMA = 0

Constant = On

Important properties of the seasonal random walk model:

• Predicted seasonal pattern = most recent seasonal pattern (last year's)

• Long-term trend in forecasts = average long-term trend in the past

• Slow to respond to cyclical upturns or downturns--always looking one year in the past

• Inherently an additive model, but can be used in conjunction with logging and/or deflating to capture multiplicative seasonal patterns

Frequent problem:

• The errors are strongly positively autocorrelated.

• This problem can often be alleviated by setting AR=1. (This adds one lag of the seasonal difference of Y to the right-hand-side of the forecasting equation. We'll discuss the rationale for this in more detail when we get to ARIMA models, but the rough rule of thumb is this: if there is a positive spike in autocorrelation function at lag 1, it often helps to set AR=1. If there is a positive spike at the seasonal period--e.g., at lag 12 for monthly data--then it often helps to set SAR=1, which adds a seasonal lag of the seasonal difference.)

If Y is strongly seasonal but the seasonal difference looks more like a random walk than noise...

...then perhaps the first difference of the seasonal difference of Y looks like noise.

Then the season-to-season trend observed last month is our best guess for the season-to-season trend that will be observed this month, i.e., Y is a seasonal random trend.

• Actual year-to-year change is assumed to be slowly and randomly changing

• Predicted year-to-year change = most recent year-to-year change:

Y(t) - Y(t-12) = Y(t-1) - Y(t-13) , or equivalently,

Y(t) = Y(t-1) + Y(t-12) - Y(t-13)

Example:

Y(Sep'96) = Y(Sep'95) + (Y(Aug'96)-Y(Aug'95))

or equivalently:

Y(Sep'96) = Y(Aug'96) + (Y(Sep'95)-Y(Aug'95))

Thus, if August's sales this year turn out to be 10,000 units ahead of August's sales last year, then we predict (at that point) that September's sales this year will also be 10,000 units ahead of September's sales last year. This model will tend to respond quickly to situations in which one year's values are consistently above or consistently below the previous year's values. (But note that it is also quick to change its mind: if September's actual sales turn out to be 5000 units less than last September's, we now predict that October's sales will also be 5000 units less than last October's, and so on.)

Seasonal Random Walk in Statgraphics:

Model type = ARIMA

Differencing:

• Nonseasonal order = 1
• Seasonal order = 1

AR, MA, SAR, SMA = 0

Constant = Off

Important properties of the seasonal random trend model:

• Predicted seasonal pattern = most recent seasonal pattern (same as seasonal random walk)

• Long-term trend in forecasts = most recent year-to-year trend

• Quick to respond to cyclical upturns or downturns--always looking at most recent trend

• Inherently an additive model, but can be used in conjunction with logging and/or deflating to capture multiplicative seasonal patterns

Frequent problems:

• The errors are strongly negatively autocorrelated (too much zigging and zagging, overcorrecting)

• The confidence intervals are huge!

• These problems can often be alleviated by setting MA=1 and/or SMA=1. (Here the rule of the thumb is that if there is a negative spike at lag 1 in the autocorrelation function, it often helps to set MA=1, which adds one lag of the forecast error to the right-hand-side of the forecasting equation. If there is a negative spike at the seasonal period, it often helps to set SMA=1, which adds a seasonal lag of the forecast error.)

A more elaborate, and more explicit, method of modeling seasonality:

Seasonal adjustment by the "ratio-to-moving average method"

Assumption: seasonal pattern is multiplicative and roughly constant from year to year

Seasonal indices can be estimated by averaging the seasonal variations (in percentage terms) that were observed in the past.

Four step process of seasonal adjustment:

(i) Compute a one-year-wide centered moving average: this is sometimes called the trend-cyclical component, and it represents a smoothed, deseasonalized estimate of the "normal" level of activity at each point in time

(ii) Compute the ratio of each period's actual value to the centered moving average in the same period: this is the percentage of "normal" observed at that point

(iii) Average the ratios computed in step (ii) for each season of the year--e.g., compute the average percentage above/below normal for January, February etc.--these are the seasonal indices

(iv) Divide each period's actual value by the seasonal index for that season of the year: this yields the seasonally adjusted data

Classical decomposition method of forecasting

Assume Y = S*T*C*I, where:

• S = seasonal component
  
• T = trend component
  
• C = cyclical component
  
• I = irregular (random error) component

Procedure:

• Use seasonal adjustment to estimate S

• Fit trend line to deseasonalized data to estimate T

• Regress deviations from trend line on a business cycle indicator variable to estimate C

• What's left over is I

Problems:

• Apparent "cyclical" behavior may be merely the artifact of a bad-fitting trend line (business cycles do exist, but even if they didn't, you would find them in your data anyway using this method)
  
• Even if you can separate the cyclical component from the trend component in past data, how do you then extrapolate it into the future?
  
• If you are going to use a commercially-provided forecast of a cyclical indicator as an input to your model, then you should use past forecasts of the indicator, rather than actual past values, when fitting regression models to estimate the cyclical component of your model--but past forecasts are often hard to obtain, and their correlations with your variable of interest are often disappointingly low.
  
• Business cycle indicators tend to be better at predicting the last recession or recovery than the next one

Modified decomposition approach:

• Use seasonal adjustment to deseasonalize the data

• Fit a nonseasonal model (e.g., random walk, smoothing or averaging) to what's left--i.e., forecast the "trend-cyclical" part as a unit

In Statgraphics:

• Choose model type = random walk or some variety of smoothing or averaging

• Choose seasonal adjustment = multiplicative

Note: you cannot combine ARIMA with seasonal adjustment in Statgraphics--ARIMA models deal with seasonality via seasonal lags and differences

Potential problem:

The seasonal adjustment process adds a large number of parameters to the model, which can lead to problems with overfitting the data. For example, when you seasonally adjust monthly data, you are estimating 12 seasonal indices along with whatever other parameters the model may have. The seasonal indices do not show up on the reports generated in the Forecasting procedure, so it is easy to forget that they are there. (You need to run the Seasonal Decomposition procedure separately in order to find out the values of the seasonal indices.) If you do not have many (say, 5 or more) seasons of data, the estimation of the seasonal indices may overfit the data. Hence, it is especially important to hold out data for validation when using seasonal adjustment. Beware of a seasonally-adjusted model that fits the data remarkably well in the estimation period and remarkably poorly in the validation period.

Revised September 7, 1997