FORECASTIN G Qualitative Analysis ~ Quantitative Analysis

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

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 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 x 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) (10) + 0.3(5) + 0.6(2) y t+1 = 10 – 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

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

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 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?

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, dollars each year over the period 1984 to Linear Trend Line: Example link to spreadsheet St = -6, ,407.3t ( ) (171.00) Note:

Linear Trend of Microsoft Corp. Sales Revenue,

Key Issue: Forecasting Annual Sales Revenue from YeartPredicted Sales , ,407.3(19) = 20, , ,407.3(20) = 21, , ,407.3(21) = 23, , ,407.3(22) = 24, , ,407.3(23) = 25, , ,407.3(24) = 27, , ,407.3(25) = 28, , ,407.3(26) = 30, , ,407.3(27) = 31,557.5

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.

Non-Linear Trend Line (Quadratic Trend): Example - Proposed Model: - Microsoft annual sales revenue ( ) * 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 = , = , n = 18 * What is the sales revenue at t = 0? * Calculate link to data = t S = – t t² (786.1) (190.5) (9.7) Standard errors in parentheses

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

Key Issue: Forecasting Annual Sales Revenue from 2002 to 2010 Yeartt²Predicted Sales – (19) (19)² = 29, – (20) (20)² = 33, – (21) (21)² = 38, – (22) (22)² = 43, – (23) (23)² = 48, – (24) (24)² = 53, – (25) (25)² = 59, – (26) (26)² = 65, – (27) (27)² = 71,558.52

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?

Exponential Trend Line: Example Proposed model: Regression model: Microsoft annual sales revenue ( ) - 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

Exponential Trend of Microsoft Corp. Sales Revenue,

Key Issue: Forecasting Annual Sales Revenue from Exponential TrendS t = 96.38*exp(0.336t) YeartPredicted Sales *exp(0.336*19) = 57, *exp(0.336*20) = 80, *exp(0.336*21) = 111, *exp(0.336*22) = 156, *exp(0.336*23) = 219, *exp(0.336*24) = 306, *exp(0.336*25) = 429, *exp(0.336*26) = 601, *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:

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.

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 Quarter 1: Jan. – March Quarter 2: April – June Quarter 3: Jul. – Sept. Quarter 4: Oct. – Dec.

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)

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

Key Issue: Forecasting Quarterly Temperature in a Resort City for 2005 and 2006 YearQuarterPredicted Temperature – = ≈ – 1.45 = ≈ = ≈ ≈ – = ≈ – 1.45 = ≈ = ≈ ≈ 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: - 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

Moving Average: Example MonthObserved S2 month MASq Err 2 MAAbs Err 2 MA3 month MASq Err 3 MAAbs Err 3 MA RMSEMADRMSEMAD RMSE MAD In this case, choose 2 mo. MA over 3 mo. MA

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

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 = MAD =

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 MAD AR(2) is the preferred model.