Managerial Economics

Principles and Worldwide

Applications, 7th Edition

Dominick Salvatore &

Ravikesh Srivastava

Chapter 5: Demand Forecasting

Qualitative Forecasts Survey Techniques

Planned Plant and Equipment Spending Expected Sales and Inventory Changes Consumers’ Expenditure Plans

Opinion Polls Business Executives Sales Force Consumer Intentions

Time-Series Analysis Secular Trend

Long-Run Increase or Decrease in Data Cyclical Fluctuations

Long-Run Cycles of Expansion and Contraction – depends in industry.

Seasonal Variation Regularly Occurring Fluctuations – sweaters.

Irregular or Random Influences – natural disasters, wars, strikes, etc.

For example, fitting a regression line to the electricity sales data running from 1Q’07 (t = 1) to the 4Q’10 (t = 16) given in the above table we get estimated regression equation of

St = 11.90 + 0.394t R2 = 0.50(4.00)

The equation indicates that electricity sales in the 4Q’06 (that is S0) is estimated to be 11.9, kw-hours and increase at the avg rate of 0.394 kw-hours per qrtr. The trend variable is statistically significant at better than the 1 percent level (inferred from the value of 4 for the t-static) and explains 50 percent quarterly variation in the city - R2 . Thus, based on past trend (of 16 quarters), we can forecast electricity consumption in (in million k-w hours) in the city to be:

S17= 11.90 + 0.394(17) = 18.60 (1Q’11) S18= 11.90 + 0.394(18) = 18.99 (2Q’11)S19= 11.90 + 0.394(19) = 19.39 (3Q’11)S20= 11.90 + 0.394(20) = 19.78 (4Q’11)

Trend Projection Linear Trend:

St = S0 + b tb = Growth per time period

Constant Growth RateSt = S0 (1 + g)t

g = Growth rate Estimation of Growth Rate

lnSt = lnS0 + t ln(1 + g)

PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.

Slide 10

Seasonal variation

The above example just considers the extended trend line and projects the long-run trend. However, during 2007 and 2010, there are strong seasonal variaitons, with sales in the first and third quarters of each year consistently below the corresponding long-run trend values, while sales in the second and fourth quarters consistently above trend values. By incorporating the seasonal variation, we can significantly improve the forecast. We do this by ratio-to-trend method or with dummy variable.

To use this method, we simply find the average ratio by which the actual value differs from estimated value during 2007-10 period and multiply the forecasted trend value by this ratio. The predicted trend value for each quarter in the 2007-10 period is obtained by simply substituting the value of t corresponding to the quarter under consideration into the given equation and solving for St. The table shows calculations for the seasonal adjustment of the sales forecast for each quarter from the extended trend line examined earlier.

Seasonal Variation

Ratio to Trend Method

ActualTrend ForecastRatio =

SeasonalAdjustment =

Average of Ratios forEach Seasonal Period

AdjustedForecast =

TrendForecast

SeasonalAdjustment

(Column Four)

(Row Average of Column Four)

Seasonal VariationExample Calculation for : Ratio to Trend Method

Multiplying the electricity sales forecasted earlier:

S17= 11.90 + 0.394(17) = 18.60 (1Q’11) S18= 11.90 + 0.394(18) = 18.99 (2Q’11)S19= 11.90 + 0.394(19) = 19.39 (3Q’11)S20= 11.90 + 0.394(20) = 19.78 (4Q’11)

By the seasonal quarterly average estimated in the table (for instance Q1 avg. = 0.887, Q2 avg. = 1.165,Q3 avg. = 0.907, Q4 avg. = 1.042, we get the following new forecasts based on both linear trend and seasonal adjustment:

S17= 18.60(0.887)= 16.50(1Q’11) S18= 18.99(1.1.65)= 22.12 (2Q’11)S19= 19.39(0.907)= 17.59 (3Q’11)S20= 19.78(1.042)= 20.61 (4Q’11)

It is important to remember that these forecasts are based on the assumption that past trend and seasonal patters in the data will persist during 2011. However, if pattern changes drastically (due to cyclic or random factors), or if we are attempting to forecast far into the future forecasts will be far off the mark. Since it cannot predict the trend – we use it along with other forecasting methods to improve our accuracy.

AdjustedForecast =

TrendForecast

SeasonalAdjustment

Moving Average Forecasts

Forecast is the average of data from w periods prior to the forecast data point.

1

wt i

ti

AF

w

Root Mean Square Error

For example, we used the three-quarter and five-quarter moving average method – to decide which of this is better, we find RMSE and use the moving avg. that results in the smallest RMSE.

That is for three-quarterly moving avg., RMSE is:

= 2.95

For five-quarter moving avg., RMSE is:

Thus, the three-quarterly moving average has a lower root-mean-square-error and so we are more confident on it.

2( )t tA FRMSE

n

Measures the Accuracy of a Forecasting Method

Exponential Smoothing Forecasts

1 (1 )t t tF wA w F

A serious criticism of the simple moving averages method is that it gives equal weight to all observations, even though we except recent observations to be more important. To overcome this, exponential smoothing gives more weight to recent periods. Here, the forecast for period t + 1 (i.e. F t+1 ) is a weighted value of the actual and forecast values of the time series in period t. The value of the time series at period t (i.e. At ) is assigned a weight (w) between 0 and 1 and the forecast for the period (i.e. Ft ) is assigned the weight of 1 – w (Note: sum of weights equals one – i.e. w + (1 – w) = 1). The greater the value of w, the greater is the weight given to the value of the time series in period t as opposed to previous periods. While Ft+1 is calculated from the value of the time series and its forecast for period t only, the forecast for period t can be shown to depend on all past values of the time series, with weights declining exponentially for values further into the past – i.e. why the name exponential smoothing. Thus the value of the forecast of the time series in period t + 1 is given below. Two decisions must be made to use this equation. 1) assign a value to initial forecast (F t) to get the analysis started. One way to do this is to let Ft equal the mean value of the entire observed time-series data. 2) We must also assign value to w (weight of A t ). In general, different values of w are tried and the one with smallest RMSE is used in forecasting.

0 1w

For example, column 3 in the table shows the forecasts for the firm’s market share given in columns 1 and 2 (same as previous example) by using the avg. mkt. share of the firm over the 12 quarters for which we have data (i.e. 21.0) for F1(to get the calculations started) and w = 0.3 as the weight for At . Thus, F2(the second value in the column is:

F2= 0.3(20) + (1 – 0.3)21 = 20.7, until F13= 21.0 for the thirteenth quarter. Similarly, for w = 0.5

1 (1 )t t tF wA w F

Exponential Smoothing Forecasts

The RMSE for the forecast using w = 0.3, is

RMSE =

The RMSE for the forecast using w = 0.5, is

RMSE =

Thus, forecast using w = 0.3 is better since it has a lower RMSE.

Barometric Methods National Bureau of Economic Research Department of Commerce Leading Indicators: leads general business activity. Lagging Indicators: lags general economic activity. Coincident Indicators –coincides with general economic activity. Composite Index: of leading, lagged and coincident indicators – weighted

average of individual indicators in each group. It measures percentage change (increase or decrease) over the previous time period with base period = 100. For eg. 89% is 11% decrease and 103% is 3% increase.

Diffusion Index: gives percentage of leading indicators moving upward or downward. If all 10 move up, the index is 100; if none move it is 0, if 7/10 move up its 70.

Econometric Models

Single Equation Model of the Demand for Cereal (Good X)

QX = a0 + a1PX + a2Y + a3N + a4PS + a5PC + a6A + e

QX = Quantity of X

PX = Price of Good X

Y = Consumer Income

N = Size of Population

PS = Price of Muffins

PC = Price of Milk

A = Advertising

e = Random Error

Econometric Models

Multiple Equation Model of GNP

1 1 1t t tC a bGNP u

2 2 1 2t t tI a b u

t t t tGNP C I G

2 11 21

1 11 1 1t t

t

b Ga aGNP b

b b

Reduced Form Equation

Chapter 5

Appendix

INTEGRATING CASE STUDY 3

Estimating and Forecasting the U.S. Demand for Electricity