Forecasting & Time Series Minggu 6

Learning Objectives Understand the three categories of forecasting techniques available. Become aware of the four components that make up a time series. Understand how to identify which components are present in a specific time series.

Learning Objectives, continued Recognize the forecasting methods available for time series with specific components. Learn several ways of identifying the forecasting methods with the least forecasting error. Forecast for time series with specific components using stationary methods, trend methods, and seasonal methods.

Introduction to Forecasting Forecasting is the art or science of predicting the future. Forecasting techniques (1) Qualitative techniques: Subjective estimates from informed sources that are used when historical data are scarce or non-existent - Examples: Delphi techniques, scenario writing, and visionary forecast.

Introduction to Forecasting, continued (2) Time Series Techniques: Quantitative techniques that use historical data for only the forecast variable to find patterns. - Based on the premise that the factors that influenced patterns of activity in the past will continue to do so in the future. -Examples: moving averages, exponential smoothing, and trend projections

Introduction to Forecasting, continued (3) Causal Techniques: Quantitative techniques based on historical data for the variable being forecast, and one or more explanatory variables. - Based on the supposition that a relationship exists between the variable to be forecast and other explanatory time series data. - Examples: regression models, econometric models, and leading indicators

Time Series Components Trend: Long-term upward or downward change in a time series Seasonal: Periodic increases or decreases that occur within one year Cyclical: Periodic increases or decreases that occur over more than a single year Irregular: Changes not attributable to the other three components; non-systematic and unpredictable

Components of Time Series Data Trend Irregular Seasonal Cyclical

Components of Time Series Data Year Seasonal Cyclical Trend Irregular fluctuations

Composite Time Series Data Year

Time Series Forecasting Procedure Step 1: Identifying Time Series Form Trend component –time series plot –trend line Seasonal component –folded annual time series plot –autocorrelation

Step 2: Select Potential Methods Stationary forecasting methods are effective for a stationary time series, that is one that contains only an irregular component. These methods attempt to eliminate the irregular through averaging. Trend forecasting methods are effective for time series that contain a trend component. These methods asses the trend component and use it to make projections. Seasonal forecasting methods are used for a time series that contains a trend, a seasonal and an irregular component.

Step 3: Evaluate Potential Methods Once the appropriate method has been chosen, it is used to forecast the historical data for the time series. The an evaluation is done of how close the estimates approach the actual historical data. Forecasting Error: A single measure of the overall error of a forecast for an entire set of data. Error of an Individual Forecast: The difference between the actual value and the forecast of that value. e t = Y t - F t

Reasons for Forecast Failure Failure to examine assumptions Limited expertise Lack of imagination Neglect of constraints Excessive optimism Reliance on mechanical extrapolation Premature closure Over specification

Measurement of Forecasting Error  Mean Error (ME): The average of all the errors of forecast for a group of data.  Mean Absolute Deviation (MAD): The mean, or average of the absolute values of the errors.  Mean Square Error (MSE): The average of the squared errors.  Mean Percentage Error (MPE): The average of the percentage errors of a forecast.  Mean Absolute Percentage Error (MAPE): The average of the absolute values of the percentage errors of a forecast.

Example: Nonfarm Partnership Tax Returns: Actual and Forecast with  =.7 YearActualForecastError

Mean Error for the Nonfarm Partnership Forecasted Data YearActualForecastError

Mean Absolute Deviation for the Nonfarm Partnership Forecasted Data YearActualForecastError|Error|

Mean Square Error for the Nonfarm Partnership Forecasted Data YearActualForecastErrorError

Mean Percentage Error for the Nonfarm Partnership Forecasted Data YearActualForecastErrorError % % % % % % % % % % % 31.8%

Mean Absolute Percentage Error for the Nonfarm Partnership Forecasted Data YearActualForecastError|Error %| % % % % % % % % % % 40.3%

Use of Error Measures To identify the best forecasting method Use error measure to identify the best value for the parameters of a specific method. Use error measure to identify the best method. Use MSE and MAD for both of these situations. Note that MSE tends to emphasize large errors.

Use of Error Measures, continued Forecast bias is the tendency of a forecasting method to over or under predict. The mean error, ME, measures the forecast bias.

Step 4: Make Required Forecasts The best forecasting method is that with the smallest overall error measurement. Using a stationary method will make a forecast for one time into the future, F t+1. This is also the forecast for all future time periods. Forecasts made using a non-stationary method will not be the same for all time periods in the future.

Stationary Forecasting Methods Naive Forecasting Method Moving Average Forecasting Method Weighted Moving Average Forecasting Method Exponential Smoothing Forecasting Method

Naive Forecasting Simplest of the naive forecasting models Simplest of the naive forecasting models We sold 532 pairs of shoes last week, I predict we’ll sell 532 pairs this week. We sold 532 pairs of shoes last week, I predict we’ll sell 532 pairs this week.

Simple Average Forecasting Method The monthly average last 12 months was 56.45, so I predict for September. The monthly average last 12 months was 56.45, so I predict for September. MonthYear Cents per GallonMonthYear Cents per Gallon January January February63.3February58.3 March62.1March57.7 April59.8April56.7 May58.4May56.8 June57.6June55.5 July55.7July53.8 August55.1August52.8 September55.7September October56.7October November57.2November December58.0December

Moving Average Forecasting Method Updated (recomputed) for every new time period May be difficult to choose optimal number of periods May not adjust for trend, cyclical, or seasonal effects Update me each period.

Weighted Moving Average Forecasting Method

Exponential Smoothing Forecasting Method  is the exponential smoothing constant

Trend Forecasting Methods Linear Trend Projection Forecasting Method: Forecasting by fitting a linear equation to a time series Non-linear Trend Projection Forecasting Method: Forecasting by fitting a non-linear equation to a time series

Average Hours Worked per Week by Canadian Manufacturing Workers PeriodHoursPeriodHoursPeriodHoursPeriodHours

Excel Regression Output using Linear Trend Regression Statistics Multiple R0.782 R Square0.611 Adjusted R Square Standard Error0.509 Observations35 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-value Intercept Period

Excel Graph of Hours Worked Data with a Linear Trend Line Time Period Work Week

Excel Regression Output using Quadratic Trend Regression Statistics Multiple R R Square0.761 Adjusted R Square0.747 Standard Error0.405 Observations35 ANOVA dfSSMSFSignificance F Regression E-10 Residual Total CoefficientsStandard Errort StatP-value Intercept E-49 Period E-07 Period E-05

Excel Graph of Hourly Data with Quadratic Trend Line Period Work Week

Exponential Smoothing with Trend Effects: Holt’s Model Holt’s Model adds consideration of a trend component to the basic exponential smoothing relation.

Trend Autoregression Method Autoregression Model with two lagged variables Autoregression Model with three lagged variables A multiple regression technique in which the independent variables are time-lagged versions of the dependent variable.

Durbin-Watson Test for Autocorrelation

Overcoming the Autocorrelation Problem Addition of Independent Variables Transforming Variables –First-differences approach –Percentage change from period to period –Use autoregression

Seasonal Forecasting Methods Seasonal Multiple Regression Forecasting Method Seasonal Autoregression Forecasting Method Winter’s Exponential Smoothing Forecasting Model Time Series Decomposition Forecasting Method

Exponential Smoothing with Trend and Seasonality: Winter’s Model

Time Series Decomposition Forecasting Method Basis for analysis is the multiplicative model Y = T · C · S · I where: T = trend component C = cyclical component S = seasonal component I = irregular component

Time Series Decomposition Determine the seasonality of the time series by computing a seasonal index for each season (each quarter, each month, and so on. Divide each time series data value by the appropriate seasonal index to deseasonalize it. Identify a trend model appropriate for the deseasonalized trend model. Forecast deseasonalized values with the trend model Multiply the deseasonalized forecasts times the appropriate seasonal index to compute the final seasonalized forecasts.

Demonstration Problem 14.6: Household Appliance Shipment Data YearQuarterShipmentsYearQuarterShipments Shipments in $1,000,000.

Demonstration Problem 14.6: Graph of Household Appliance Shipment Data Quarter Shipments