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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 -Expert Opinion (Personal Insight) -Panel Consensus -Delphi method •using forecasts derived from independent analysis of expert opinion Forecasting with Multiple Regression Confidence intervals for prediction. y =b +bx + bx +bx +bx + u t 0 1 1t 2 2t 3 3t 4 4t t - Suppose that ˆ y t = 10 – 0.5x + 0.25x + 0.3x + 0.6x 1t 2t 3t 4t - Provide a forecast for y t+1 - To do so, we need future values of x , x , x , and x 1t 2t 3t 4t Suppose that: x = 12 1t+1 x = 10 2t+1 x =5 x =2 3t+1 4t+1 Then y = 10 – (0.5)(12) + 0.25(10) + 0.3(5) + 0.6(2) t+1 y = 10 – 6 + 2.5 + 1.5 + 1.2 t+1 y = 9.2 t+1 The forecast is conditional upon future values of x , x , x , and x . 1t 2t 3t 4t This forecast is a point forecast Confidence Interval for Prediction (or Forecast) with Multiple Regression This confidence interval is given by: point forecast ± se(regression) * critical value c se(regression) = residual variance critical value c: t , α n-p Suppose that se(regression) = 2.4 and that t , α = 0.05 = 1.8 n-5 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] In general, any time series may be decomposed into four components: 1. trend component 2. seasonal component 3. cyclical component 4. random component 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 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 Time-Series Characteristics: Secular Trend and Cyclical Variation in Women’s Clothing Sales Time-Series Characteristics: Seasonal Pattern and Random Fluctuations Linear Trend yt 0 1 t t y : variable of interest t t: time, t = 1, 2, …, T ß : intercept 0 Questions: ß : slope, a constant change in 1 -Does a linear trend have any the series from one curvature? period to the next period -How to interpret ß ? 0 -If ß > 0, what does it mean? 1 -If ß < 0, what does is mean? 1 Linear Trend Line: Example Proposed model: S = a + b + ε t t t -Microsoft annual sales revenue (1984 – 2001) link to spreadsheet * 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 dS * conduct an OLS regression Note: b dt -Estimation results: St = -6,440.8 + 1,407.3t -Question: (1850.96) (171.00) * What is the sales revenue at t = 0? ˆ * interpret b 1,407 .3 The series grows by $1,407.30 dollars each year over the period 1984 to 2001. Linear Trend of Microsoft Corp. Sales Revenue, 1984-2001 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 Non-Linear Trend: Quadratic Trend yt 0 1 t 2 t t 2 y : variable of interest t t: time, t = 1, 2, …, T ß : intercept 0 Questions: Marginal increase from this - Does a quadratic trend have period to the next one: any curvature? dy 1 2 2t - How does the series grow dt (or decline) each period? Calculate dy Note: this growth dt or decline depends on t. Non-Linear Trend Line (Quadratic Trend): Example - Proposed Model: - Microsoft annual sales revenue (1984-2001) link to data * 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 S = 2628.7 – 1313.5t + 143.2t² (786.1) (190.5) (9.7) Standard errors in parentheses 2 - Question: R² = 0.9876, R = 0.9860, n = 18 ds * What is the sales revenue at t = 0? = -1313.5 + 286.4t dt * Calculate ds dt Non-Linear Trend – Quadratic Trend of Microsoft Corp. Sales Revenue, 1984- 2001 S 2628.7 1313.5t 143.20t 2 $30,000 $25,000 Sale revenue (per million $) $20,000 $15,000 $10,000 $5,000 $0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 -$5,000 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 Exponential Trend t yt exp 1 expt ln( yt ) 0 1 t t where 0 ln( ) Note : exp1t e1t and expt et y : variable of interest t t: time, t = 1, 2, …, T ß : intercept 0 Questions: The series grows (if ß > 0) or 1 - Does an exponential trend declines (if ß < 0) by a constant 1 have any curvature? percentage. - If ß > 0, what does this 1 finding mean? - If ß < 0, what does this 1 finding mean? Exponential Trend Line: Example 1t Proposed model: S t exp Regression model: ln( S t ) 0 1 t link to data Microsoft annual sales revenue (1984-2001) - S = annual sales revenue - t = time period ˆ - estimation of α: 0 ln( ) exp 0 ˆ How to approach? - create two additional columns *Log(S1) = log(sales revenue) * t for time period Estimation results ln( St ) 4.568 0.336t (0.12) (0.01) R 0.9831, adjusted R 0.9821, n 18 2 2 Questions: - What is the sales revenue at t = 0? ( exp 4.568 96 .38 ) ˆ ˆ - By what constant percentage does sales revenue grow? 1 0.336 The series grows by 33.6% each year. Exponential Trend of Microsoft Corp. Sales Revenue, 1984-2001 ˆ ˆ ln( y t ) 4.568 0.336 t 0 1 t t where 0 ln( ). ˆ ˆ yt exp 1 96.38 exp ˆ ˆ 0.336t ˆ Thus, exp 0 exp 4.59 96.38. ˆ $45,000 $40,000 Sale revenue (per million $) $35,000 $30,000 $25,000 $20,000 $15,000 $10,000 $5,000 $0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Key Issue: Forecasting Annual Sales Revenue from 2002-2010 Exponential Trend S = 96.38*exp(0.336t) t 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 Seasonal Variation Common Examples: - Christmas shopping rush - seasonal products and activities (Halloween candy, Thanksgiving turkey) - weekends vs. weekdays - sports seasons and events - political elections 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: 1 if 1st quarter 1 if 2 nd quarter 1 if 3rd quarter 1 if 4 th quarter D1 D2 D3 D4 0 otherwise 0 otherwise 0 otherwise 0 otherwise 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. S a c1 D1 c2 D2 c3 D3 - In a comparison with the fourth quarter (D is the base), sales 4 change by c in the first quarter, c in the second quarter, and c in the 1 2 3 third quarter. Seasonal Dummy: Example Quarterly Temperature Readings in a Resort City Over the Period 1994 to 2004 TEMPERATURE Quarter 1: Jan. – March Quarter 2: April – June Year Quarter Temperature 90 Quarter 3: Jul. – Sept. 1994 1 47 Quarter 4: Oct. – Dec. 1994 2 65 80 1994 3 83 1994 4 67 1995 1 51 70 1995 2 64 1995 3 82 1995 4 66 60 . . . . . . 50 . . . 2004 1 48 2004 2 67 40 2004 3 80 94 95 96 97 98 99 00 01 02 03 04 2004 4 67 Note the Regular Periodicities of the Temperature Data Seasonal Dummies: Example continued . . . Define dummy variables: 1 if 1st quarter 1 if 2 nd quarter 1 if 3rd quarter D 1 if 4 th quarter D1 D2 D3 4 0 otherwise 0 otherwise 0 otherwise 0 otherwise Regression Model: TEMP a c1 D1 c2 D2 c3 D3 - Why is the 4th quarter (D ) omitted? (Base Case) 4 - Does it matter if we use another base? (No) Seasonal Dummies: Example continued . . . Regression Results: temp 66 .84 17 .64 D1 1.45 D2 16 .27 D3 ˆ (0.48) (0.68) (0.68) (0.68) n² = 0.9841 R² = 0.9829, n = 44 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 Smoothing Techniques - Take into account cyclical components in a time-series. - Smoothing Techniques: Moving Average model Autoregressive model Moving Average (MA) Forecasts - N-period MA forecasts the next period as the average of the last N periods: 1 S t 1 S t S t 1 ... S t N 1 ˆ N - 3-month MA projection of sales for March is average sales in Feb., Jan., and Dec. ˆMar 1 S Feb S Jan S Dec S 3 - 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 Moving Average: Example Month Observed S 2 month MA Sq Err 2 MA Abs Err 2 MA 3 month MA Sq Err 3 MA Abs Err 3 MA 1 1100 2 1891 3 1769 1495.5 74802.25 273.5 4 1897 1830 4489 67 1586.67 96306.78 310.33 5 1798 1833 1225 35 1852.33 2952.11 54.33 6 2168 1847.5 102720.25 320.5 1821.33 120177.78 346.67 7 2364 1983 145161 381 1954.33 167826.78 409.67 8 2554 2266 82944 288 2110.00 197136.00 444.00 9 3387 2459 861184 928 2362.00 1050625.00 1025.00 10 2079 2970.5 794772.25 891.5 2768.33 475180.44 689.33 11 2890 2733 24649 157 2673.33 46944.44 216.67 12 2690 2484.5 42230.25 205.5 2785.33 9088.44 95.33 RMSE MAD RMSE MAD RMSE MAD RMSE MAD 462.0 354.7 490.6 399.04 n S n S MAt MAt 2 t t MSE t 1 MAD t 1 n n In this case, choose 2 mo. MA over 3 mo. MA RMSE MSE Autoregressive (AR) Model - Time-series approach, univariate model - Autoregressive model of order 1: AR(1) yt 0 1 yt 1 t - Autoregressive model of order p: AR(p) yt 0 1 yt 1 2 yt 2 p yt p t - How to approach? OLS regressions Autoregressive Model: AR(2) Create two variables, S and S t-1 t-2 link for data 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 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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