1. FORECASTIN G Qualitative Analysis ~ Quantitative Analysis

2. -Multiple Regression -Confidence Interval for Prediction -Trend Analysis and Projections -Seasonal Models -Smoothing Techniques Predictions or Forecasting with:

3. Qualitative Analysis -Surveys -Polling -Expert Opinion (Personal Insight) -Panel Consensus -Delphi method using forecasts derived from independent analysis of expert opinion

4. Forecasting with Multiple Regression Confidence intervals for prediction. y t = b 0 + b 1 x 1t + b 2 x 2t + b 3 x 3t + b 4 x 4t + u t - Suppose that= 10 – 0.5x 1t + 0.25x 2t + 0.3x 3t + 0.6x 4t - Provide a forecast for y t+1 - To do so, we need future values of x 1t, x 2t, x 3t, and x 4t Suppose that: x 1t+1 = 12x 2t+1 = 10x 3t+1 = 5x 4t+1 = 2 Then y t+1 = 10 – (0.5)(12) + 0.25(10) + 0.3(5) + 0.6(2) y t+1 = 10 – 6 + 2.5 + 1.5 + 1.2 y t+1 = 9.2 The forecast is conditional upon future values of x 1t, x 2t, x 3t, and x 4t. This forecast is a point forecast

5. This confidence interval is given by: point forecast ± se(regression) * critical value c se(regression) = critical value c: t n-p, α Suppose that se(regression) = 2.4 and that t n-5, α = 0.05 = 1.8 With our point forecast of 9.2, then the 95% confidence interval for prediction is given by: 9.2 ± (2.4)(1.8) 9.2 ± 4.32[4.88, 13.52] Confidence Interval for Prediction (or Forecast) with Multiple Regression

6. In general, any time series may be decomposed into four components: 1.trend component 2.seasonal component 3.cyclical component 4.random component

7. Time-Series Analysis of Forecasting Develop models to stress trend component, seasonal component, and cyclical components. -trend analysis and projection -seasonal models -smoothing techniques (cyclical components) Moving Average Models Autoregressive Models

8. Trend Analysis and Projections Trend Analysis - forecast the future path of economic variables based on historical data - use a regression model to model the trend as a function of time Types of trend analysis - linear trend - nonlinear trend - seasonal variations

9. Time-Series Characteristics: Secular Trend and Cyclical Variation in Women’s Clothing Sales

10. Time-Series Characteristics: Seasonal Pattern and Random Fluctuations

11. Linear Trend y t : variable of interest t: time, t = 1, 2, …, T ß 0 : intercept ß 1 : slope, a constant change in the series from one periodto the next period Questions: -Does a linear trend have any curvature? -How to interpret ß 0 ? -If ß 1 > 0, what does it mean? -If ß 1 < 0, what does is mean?

12. Proposed model: S t = a + b t + ε t -Microsoft annual sales revenue (1984 – 2001) * S = annual sales revenue * t = time period * a = sales revenue at t = 0 (may or may not be meaningful) * b = series grows ( if b > 0) or declines (if b < 0) by a constant amount -How to conduct a linear trend analysis? * create another column for t * conduct an OLS regression -Estimation results: -Question: * What is the sales revenue at t = 0? * interpret The series grows by $1,407.30 dollars each year over the period 1984 to 2001. Linear Trend Line: Example link to spreadsheet St = -6,440.8 + 1,407.3t (1850.96) (171.00) Note:

13. Linear Trend of Microsoft Corp. Sales Revenue, 1984-2001

14. Key Issue: Forecasting Annual Sales Revenue from 2002-2010 YeartPredicted Sales 200219-6,440.8 + 1,407.3(19) = 20,298.7 200320-6,440.8 + 1,407.3(20) = 21,706.1 200421-6,440.8 + 1,407.3(21) = 23,113.4 200522-6,440.8 + 1,407.3(22) = 24,520.8 200623-6,440.8 + 1,407.3(23) = 25,928.1 200724-6,440.8 + 1,407.3(24) = 27,335.5 200825-6,440.8 + 1,407.3(25) = 28,742.8 200926 -6,440.8 + 1,407.3(26) = 30,150.1 201027-6,440.8 + 1,407.3(27) = 31,557.5

15. y t : variable of interest t: time, t = 1, 2, …, T ß 0 : intercept Marginal increase from this period to the next one: Non-Linear Trend: Quadratic Trend Questions: - Does a quadratic trend have any curvature? - How does the series grow (or decline) each period? Calculate Note: this growth or decline depends on t.

16. Non-Linear Trend Line (Quadratic Trend): Example - Proposed Model: - Microsoft annual sales revenue (1984-2001) * S = annual sales revenue * t = time period * a = sales revenue at t = 0 (may or may not be meaningful) * b 1 and b 2 : trend parameters - How to approach? * create two additional columns * conduct an OLS regression - Estimation Results - Question: R² = 0.9876, = 0.9860, n = 18 * What is the sales revenue at t = 0? * Calculate link to data = -1313.5 + 286.4t S = 2628.7 – 1313.5t + 143.2t² (786.1) (190.5) (9.7) Standard errors in parentheses

17. Non-Linear Trend – Quadratic Trend of Microsoft Corp. Sales Revenue, 1984-2001

18. Key Issue: Forecasting Annual Sales Revenue from 2002 to 2010 Yeartt²Predicted Sales 2002193612628.7 – 1313.5(19) + 143.2(19)² = 29,368.19 2003204002628.7 – 1313.5(20) + 143.2(20)² = 33,639.57 2004214442628.7 – 1313.5(21) + 143.2(21)² = 38,197.35 200522484 2628.7 – 1313.5(22) + 143.2(22)² = 43,041.54 2006235292628.7 – 1313.5(23) + 143.2(23)² = 48,172.13 2007246252628.7 – 1313.5(24) + 143.2(24)² = 53,589.12 2008256252628.7 – 1313.5(25) + 143.2(25)² = 59,292.52 2009266762628.7 – 1313.5(26) + 143.2(26)² = 65,282.32 2010277292628.7 – 1313.5(27) + 143.2(27)² = 71,558.52

19. Exponential Trend y t : variable of interest t: time, t = 1, 2, …, T ß 0 : intercept The series grows (if ß 1 > 0) or declines (if ß 1 < 0) by a constant percentage. Questions: - Does an exponential trend have any curvature? - If ß 1 > 0, what does this finding mean? - If ß 1 < 0, what does this finding mean?

20. Exponential Trend Line: Example Proposed model: Regression model: Microsoft annual sales revenue (1984-2001) - S = annual sales revenue - t = time period - estimation of α: How to approach? - create two additional columns *Log(S1) = log(sales revenue) * t for time period Estimation results Questions: - What is the sales revenue at t = 0? - By what constant percentage does sales revenue grow? The series grows by 33.6% each year. link to data

21. Exponential Trend of Microsoft Corp. Sales Revenue, 1984-2001

22. Key Issue: Forecasting Annual Sales Revenue from 2002-2010 Exponential TrendS t = 96.38*exp(0.336t) YeartPredicted Sales 20021996.38*exp(0.336*19) = 57,182.8 20032096.38*exp(0.336*20) = 80,026.6 20042196.38*exp(0.336*21) = 111,996.0 200522 96.38*exp(0.336*22) = 156,736.9 20062396.38*exp(0.336*23) = 219,351.1 20072496.38*exp(0.336*24) = 306,978.8 20082596.38*exp(0.336*25) = 429,612.5 20092696.38*exp(0.336*26) = 601,236.6 20102796.38*exp(0.336*27) = 841,422.2

23. Seasonal Variation Common Examples: - Christmas shopping rush - seasonal products and activities (Halloween candy, Thanksgiving turkey) - weekends vs. weekdays - sports seasons and events - political elections

24. Seasonal Variation continued... Use of indicator variables or dummy variables. A dummy variable equals one when a condition is met and it equals zero otherwise. - Example: Define quarterly dummy variables as follows:

25. Seasonal Variation continued... - Run a regression with dummy variables to account for seasonality. - Note: You must leave out one of the dummy variables! Why? Perfect collinearity Which one to drop? It doesn’t matter. It will not change your R² or F statistic, coefficient estimates, or their t- statistics. How to interpret? The dummy variable left our becomes the base case. The estimated dummy coefficients are adjustments relative to this base case. - In a comparison with the fourth quarter (D4 is the base), sales change by c1 in the first quarter, c2 in the second quarter, and c3 in the third quarter.

26. Seasonal Dummy: Example Quarterly Temperature Readings in a Resort City Over the Period 1994 to 2004 Note the Regular Periodicities of the Temperature Data Year Quarter Temperature 1994 1 47 1994 2 65 1994 3 83 1994 4 67 1995 1 51 1995 2 64 1995 3 82 1995 4 66... 2004 1 48 2004 2 67 2004 3 80 2004 4 67 Quarter 1: Jan. – March Quarter 2: April – June Quarter 3: Jul. – Sept. Quarter 4: Oct. – Dec.

27. Seasonal Dummies: Example continued... Define dummy variables: Regression Model: - Why is the 4 th quarter (D 4 ) omitted? (Base Case) - Does it matter if we use another base? (No)

28. Seasonal Dummies: Example continued... Regression Results: (0.48) (0.68)(0.68) (0.68) n² = 0.9841R² = 0.9829, n = 44

29. Key Issue: Forecasting Quarterly Temperature in a Resort City for 2005 and 2006 YearQuarterPredicted Temperature 2005166.54 – 17.64 = 48.90 ≈ 49 2005266.54 – 1.45 = 65.09 ≈ 65 2005366.54 + 16.27 = 82.81 ≈ 83 2005466.54 ≈ 67 2006166.54 – 17.64 = 48.90 ≈ 49 2006266.54 – 1.45 = 65.09 ≈ 65 2006366.54+ 16.27 = 82.81 ≈ 83 2006466.54 ≈ 67

30. Smoothing Techniques - Take into account cyclical components in a time-series. - Smoothing Techniques: Moving Average model Autoregressive model

31. Moving Average (MA) Forecasts - N-period MA forecasts the next period as the average of the last N periods: - 3-month MA projection of sales for March is average sales in Feb., Jan., and Dec. - The longer the MA, the greater the smoothing: a 5-month MA is smoother than a 3-month MA - Use a longer MA when random fluctuations are a larger component of the time series. - Use RMSE and MAD to decide upon the appropriate smoothing time frame. RMSE = root mean square error MAD = mean absolute deviation

32. Moving Average: Example MonthObserved S2 month MASq Err 2 MAAbs Err 2 MA3 month MASq Err 3 MAAbs Err 3 MA 11100 21891 317691495.574802.25273.5 4189718304489671586.6796306.78310.33 5179818331225351852.332952.1154.33 621681847.5102720.25320.51821.33120177.78346.67 7236419831451613811954.33167826.78409.67 825542266829442882110.00197136.00444.00 9338724598611849282362.001050625.001025.00 1020792970.5794772.25891.52768.33475180.44689.33 1128902733246491572673.3346944.44216.67 1226902484.542230.25205.52785.339088.4495.33 RMSEMADRMSEMAD 462.0354.7490.6399.04 RMSE MAD In this case, choose 2 mo. MA over 3 mo. MA

33. Autoregressive (AR) Model - Time-series approach, univariate model - Autoregressive model of order 1: AR(1) - Autoregressive model of order p: AR(p) - How to approach? OLS regressions

34. Autoregressive Model: AR(2) link for data Create two variables, S t-1 and S t-2 Run an OLS regression - data: Months 3-12 Arrange the following values for each observation: - actual sales - predicted sales - square of error Calculate RMSE or MAD - RMSE = 387.07 - MAD = 288.62

35. Which Model is Better, MA(2), MA(3), or AR(2)? The one with the lowest RMSE or MAD. MA(2)MA(3)AR(2) RMSE 462.0490.6387.07 MAD 354.7399.04288.62 AR(2) is the preferred model.