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FORECASTING

FORECASTING. Qualitative Analysis ~ Quantitative Analysis. Predictions or Forecasting with:. Multiple Regression Confidence Interval for Prediction Trend Analysis and Projections Seasonal Models Smoothing Techniques. Qualitative Analysis. Surveys Polling

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FORECASTING

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  1. FORECASTING Qualitative Analysis ~ Quantitative Analysis

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

  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. yt = b0 + b1x1t + b2x2t + b3x3t + b4x4t + ut - Suppose that = 10 – 0.5x1t + 0.25x2t + 0.3x3t + 0.6x4t - Provide a forecast for yt+1 - To do so, we need future values of x1t, x2t, x3t, and x4t Suppose that: x1t+1 = 12 x2t+1 = 10 x3t+1 = 5 x4t+1 = 2 Then yt+1 = 10 – (0.5)(12) + 0.25(10) + 0.3(5) + 0.6(2) yt+1 = 10 – 6 + 2.5 + 1.5 + 1.2 yt+1 = 9.2 The forecast is conditional upon future values of x1t, x2t, x3t, and x4t. This forecast is a point forecast

  5. Confidence Interval for Prediction (or Forecast) with Multiple Regression This confidence interval is given by: point forecast ± se(regression) * critical value c se(regression) = critical value c: tn-p, α Suppose that se(regression) = 2.4 and that tn-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]

  6. In general, any time series may be decomposed into four components: • trend component • seasonal component • cyclical component • 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 yt: variable of interest t: time, t = 1, 2, …, T ß0: intercept ß1: slope, a constant change in the series from one period to the next period Questions: • Does a linear trend have any curvature? • How to interpret ß0? • If ß1 > 0, what does it mean? • If ß1 < 0, what does is mean?

  12. Linear Trend Line: Example Proposed model: St = a + bt + εt • 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. link to spreadsheet Note: St = -6,440.8 + 1,407.3t (1850.96) (171.00)

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

  14. Key Issue: Forecasting Annual Sales Revenue from 2002-2010 Year t Predicted Sales 2002 19 -6,440.8 + 1,407.3(19) = 20,298.7 2003 20 -6,440.8 + 1,407.3(20) = 21,706.1 2004 21 -6,440.8 + 1,407.3(21) = 23,113.4 2005 22 -6,440.8 + 1,407.3(22) = 24,520.8 2006 23 -6,440.8 + 1,407.3(23) = 25,928.1 2007 24 -6,440.8 + 1,407.3(24) = 27,335.5 2008 25 -6,440.8 + 1,407.3(25) = 28,742.8 2009 26 -6,440.8 + 1,407.3(26) = 30,150.1 2010 27 -6,440.8 + 1,407.3(27) = 31,557.5

  15. Non-Linear Trend: Quadratic Trend yt: variable of interest t: time, t = 1, 2, …, T ß0: intercept Marginal increase from this period to the next one: 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) * b1 and b2: 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 S = 2628.7 – 1313.5t + 143.2t² (786.1) (190.5) (9.7) Standard errors in parentheses = -1313.5 + 286.4t

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

  18. Key Issue: Forecasting Annual Sales Revenue from 2002 to 2010 Year t t² Predicted Sales 2002 19 361 2628.7 – 1313.5(19) + 143.2(19)² = 29,368.19 2003 20 400 2628.7 – 1313.5(20) + 143.2(20)² = 33,639.57 2004 21 444 2628.7 – 1313.5(21) + 143.2(21)² = 38,197.35 2005 22 484 2628.7 – 1313.5(22) + 143.2(22)² = 43,041.54 2006 23 529 2628.7 – 1313.5(23) + 143.2(23)² = 48,172.13 2007 24 625 2628.7 – 1313.5(24) + 143.2(24)² = 53,589.12 2008 25 625 2628.7 – 1313.5(25) + 143.2(25)² = 59,292.52 2009 26 676 2628.7 – 1313.5(26) + 143.2(26)² = 65,282.32 2010 27 729 2628.7 – 1313.5(27) + 143.2(27)² = 71,558.52

  19. Exponential Trend yt: variable of interest t: time, t = 1, 2, …, T ß0: intercept The series grows (if ß1> 0) or declines (if ß1< 0) by a constant percentage. Questions: - Does an exponential trend have any curvature? - If ß1> 0, what does this finding mean? - If ß1 < 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 Trend St = 96.38*exp(0.336t) Year t Predicted Sales 2002 19 96.38*exp(0.336*19) = 57,182.8 2003 20 96.38*exp(0.336*20) = 80,026.6 2004 21 96.38*exp(0.336*21) = 111,996.0 2005 22 96.38*exp(0.336*22) = 156,736.9 2006 23 96.38*exp(0.336*23) = 219,351.1 2007 24 96.38*exp(0.336*24) = 306,978.8 2008 25 96.38*exp(0.336*25) = 429,612.5 2009 26 96.38*exp(0.336*26) = 601,236.6 2010 27 96.38*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 Quarter 1: Jan. – March Quarter 2: April – June Quarter 3: Jul. – Sept. Quarter 4: Oct. – Dec. 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 Note the Regular Periodicities of the Temperature Data

  27. Seasonal Dummies: Example continued . . . Define dummy variables: Regression Model: - Why is the 4th quarter (D4) 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 = 0.9841 R² = 0.9829, n = 44

  29. Key Issue: Forecasting Quarterly Temperature in a Resort City for 2005 and 2006 Year Quarter Predicted Temperature 2005 1 66.54 – 17.64 = 48.90 ≈ 49 2005 2 66.54 – 1.45 = 65.09 ≈ 65 2005 3 66.54 + 16.27 = 82.81 ≈ 83 2005 4 66.54 ≈ 67 2006 1 66.54 – 17.64 = 48.90 ≈ 49 2006 2 66.54 – 1.45 = 65.09 ≈ 65 2006 3 66.54+ 16.27 = 82.81 ≈ 83 2006 4 66.54 ≈ 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 RMSE MAD 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) Create two variables, St-1 and St-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 link for data

  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 462.0 490.6 387.07 MAD 354.7 399.04 288.62 AR(2) is the preferred model.

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