Forecasting Models

I. Introduction

How many units of a product should a company produce to meet demand for the next time period? How many bushels of corn will an acre yield? What will the interest rate be? How many waiters and waitresses should be scheduled for a shift? How will the value of a stock change? What will raw material prices be? How many bottles of beer will a restaurant sell?

How do businesses answer these types of questions? Obviously, there is uncertainty associated with these scenarios, so they cannot simply calculate the answers. Rather, they forecast the answers to such questions. Business managers rely on forecasting techniques to recognize trends and develop appropriate strategies.

There are basically three types of forecasting methods:

1. Judgmental Methods – subjective, qualitative.

2. Time Series Methods (a.k.a. extrapolation methods) – quantitative, use a series of past data points to predict the future. Don’t care about what caused the past - just expect the pattern to continue.

3. Causal Methods – quantitative, forecast the quantity of interest by relating it directly to one or more other quantities.

Typically a quantitative method is used in conjunction with a judgmental method. Our focus will be on the quantitative methods, but recognize no method can accurately forecast the future every time.

II. Time Series Methods

A time series is a series of observations collected over evenly spaced intervals of some quantity of interest. e.g. number of phone calls per hour, number of cars per day, number of students per semester,…

Let

y_i = observed value i of a time series (i = 1,2,…,t)

yhat_i = forecasted value of y_i

e_i = error for case i = y_i – yhat_i

Sometimes the forecast is too high (negative error) and sometimes it is too low (positive error).

The accuracy of the forecasting method is measured by the forecasting errors. There are two popular methods for assessing forecasting accuracy:

Mean Absolute Deviation (MAD)

MAD = ( å | e_i | )/n

Where n is the number of periods in the forecast

Units of MAD are same as units of y_i

Mean Squared Error

MSE = (å e_i ² )/n

Units squared

A. Moving Averages

This is the simplest and most commonly used forecasting method. It works best when the time series fluctuates around a constant base level.

Use an average of the N most recent observations to forecast the next period. From one period to the next, the average “moves” by replacing the oldest observation in the average with the most recent observation. In the process, short-term irregularities in the data series are “smoothed out”.

MA = å y_i / N

See milk.xls for an example.

How do you choose N? Want the value of N that gives us the best forecasting accuracy. (i.e. minimizes MAD or MSE)

If the errors are normally distributed, then the standard deviation of the forecast errors (s_e) is approximately equal to 1.25 * MAD.

68% of the forecasts will be within +/- one s_e of the observed value.

95% of the forecasts will be within +/- two s_e of the observed value.

To graph the observed values and the forecasted values over time in Excel,

Highlight the observed values and the forecasted values b6:c26

Click on the chart wizard icon

Chart type: line

Next

Next

Category (X) axis: days

Value (Y) axis: gallons of milk

Next

As new sheet: Milk sales graph

Finish

B. Weighted Moving Averages

In the moving averages method, each observation in the average is equally weighted. In the weighted moving averages method, typically the most recent observation carries the most weight in the average.

Let w_i = weight for observation i

The sum of the weights is 1.

WMA = å w_iy_i

To illustrate, let’s rework the milk example using a weight of .5 for the most recent observation, a weight of .3 for the next most recent observation, and a weight of .2 for the oldest observation.

Could we have used a better set of weights than those above? Let the Excel feature Solver check for you. Choose Tools, Solver from the menu.

Target Cell is MAD

Min

Changing Cells are the weights

Constraints are that the weights must <= 1, >= 0, and add up to 1.