Chapter 6 Business and Economic Forecasting

Root-mean-squared Forecast Error zUsed to determine how reliable a forecasting technique is. zE = (Y i - F i ) 2 / n where : F i = ith forecast Y i = the corresponding actual value n = the number of forecasts i=1 n

Taking Apart a Time Series Trend: A relatively smooth long-term movement of a time-series. The value of a variable might differ from trend because of: zSeasonal variation zCyclical variation zIrregular variation

Estimating a Linear Trend Y t = A + Bt where Y t is the trend value of the variable at time t.

Estimating a Nonlinear Trend Quadratic function Y t = A + B 1 t + B 2 t 2 Exponential function Y t =  t or log Y t = log a + log b x t

Accounting for Seasonal Variation Seasonal index zDescribes the seasonal variation in a particular time series zShows the way in which that month tends to depart from what would be expected on the basis of the trend and cyclical variation in the time series

Accounting for Cyclical Variation Business cycle: describes fluctuations in the level of economic activity over time Time Level of economic activity Trough Peak Expansion Contraction

Elementary Forecasting zFundamental forecasting equation: Y t =T x S x C x I Trend Seasonal Effect Cyclical Effect Irregular Effect

Linear Trend zShows the simple, linear effects of time on the dependent variable: Y t = a + bt + e t Where t is our time index and e t is our forecasting error.

Using OLS to Estimate a Linear Trend

Example: Fitting a Linear Trend (continued) zThe forecasted equation is: thus, to forecast the 30th period we insert 30 in place of t, zNotice that the errors seem not to be random, but we witness strings of positive errors and strings of negative errors. This is typically indicative of two possible problems: autorcorrelation or mis-specified functional form. We can consider the latter by plotting the data and looking for a non-linear pattern in the data.

Non-Linear Relationships zAs the graph reveals, the data do not seem to follow a linear trend, but rather a non-linear trend:

Exponential and Polynomial Trends zTwo common non-linear trend relationships: Exponential: Y t =  t Polynomial: Y t = a + b 1 t + b 2 t 2 (quadratic form)

Exponential Trend zIn order to estimate the exponential trend, we first transform the model into a linear one by taking natural logs (and adding a stochastic error term): ln(Y t )= ln  + ln  (t) + e t By “linearizing” the exponential function we can now estimate the natural log version using OLS. Our dependent variable is no longer Sales, but the natural log of sales.

Exponential Trend (continued)

zThe estimate equation is: thus the forecasted sales for the 30th. period would be: ( A simpler way to estimate this trend is to use the Excel Chart option and add an exponential trend line.)

Quadratic Trend zIn a similar way, we can estimate an quadratic trend: or, for the 30th. period: (Done using Excel’s Chart Option to add a polynomial trend line of order 2)

Dummy Variables A variable that can assume only two values: 0 or 1

Seasonal Adjustments Using Dummy Variables zOne method of controlling for Season variation is to create seasonal dummy variables. Dummy variables, (also known as indicator or categorical variables), are simply variables that are created to indicate whether something is true. For example, Y t = a + b 1 t + b 2 D 2t + b 3 D 3t + b 4 D 4t +e t Y t = monthly sales t = time index D 2t = 1 is the month belongs to the 2nd quarter, 0 otherwise D 3t = 1 is the month belongs to the 3rd quarter, 0 otherwise D 4t = 1 is the month belongs to the 4th quarter, 0 otherwise

Using Dummy Variables (continued) zNotice that we do not include a dummy for the first quarter. This is because doing so would be redundant since if D2=0, D3=0 and D4=0, then it must be the first quarter. Thus no separate dummy variable for the first quarter is needed and the first quarter becomes our base period.

Using Dummy Variables (continued) z OLS may be applied to this model and the dummy variables are treated as any other independent variable in the regression. The adjusted R squared increases substantially (from without the dummies to with them) indicating a better fit.

Using Dummy Variables (continued) zThe predicted equation is, Predicting the 16th. month’s sales, given it is a second quarter observation, we have:

Exponential Smoothing zAnother method of forecasting values of a variable is to use a weighted average of previous values. This is precisely what exponential smoothing does. The “exponential” part of exponential smoothing refers to how the weights are assigned to previous values. The weights are assigned such that they decline exponentially as we move backward in time.

Exponential Smoothing (continued) zMathematically, let y t be our variable we wish to forecast. Then we have: zThe value of y t with the bar above is the weighted average of the previous values of y t. The parameter , is called the smoothing constant and takes on values in the interval: (0    1). Values for  close to 0 give less weight to recent values and more weight to past. Values close to 1 give greater weight to recent values and less weight to past ones.

Exponential Smoothing (continued) The steps for forecasting go as the following; 1. Initialize: 2. Update: 3. Forecast:

Exponential Smoothing (continued) Example: Suppose we have 5 years of sales data ($ millions), Let  = 0.3:

Exponential Smoothing (continued) z Excel is capable of calculating exponentially smoothed values for a given set of values. The function is found under the “Data Analysis” option under the “Tools” main menu item.

Using Economic Indicators zLeading indicators -- variables that go down before the peak and up before the trough zCoincident series -- variables that go down at the peak and up at the trough zLagging series -- variables that go down after the peak and up after the trough