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 x2t + 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) (10) + 0.3(5) + 0.6(2)
yt+1 = 10 –
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 ± [4.88, 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
- 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, dollars each year over the period 1984 to 2001.
link to spreadsheet
Note:
St = -6, ,407.3t
( ) (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
, ,407.3(19) = 20,298.7
, ,407.3(20) = 21,706.1
, ,407.3(21) = 23,113.4
, ,407.3(22) = 24,520.8
, ,407.3(23) = 25,928.1
, ,407.3(24) = 27,335.5
, ,407.3(25) = 28,742.8
, ,407.3(26) = 30,150.1
, ,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 ( )
* 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 = , = , n = 18
* What is the sales revenue at t = 0?
* Calculate
link to data
S = – t t²
(786.1) (190.5) (9.7)
Standard errors in parentheses
= t

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

18. Key Issue: Forecasting Annual Sales Revenue from 2002 to 2010
Year t t² Predicted Sales
– (19) (19)² = 29,368.19
– (20) (20)² = 33,639.57
– (21) (21)² = 38,197.35
– (22) (22)² = 43,041.54
– (23) (23)² = 48,172.13
– (24) (24)² = 53,589.12
– (25) (25)² = 59,292.52
– (26) (26)² = 65,282.32
– (27) (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 ( )
- 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
*exp(0.336*19) = 57,182.8
*exp(0.336*20) = 80,026.6
*exp(0.336*21) = 111,996.0
*exp(0.336*22) = 156,736.9
*exp(0.336*23) = 219,351.1
*exp(0.336*24) = 306,978.8
*exp(0.336*25) = 429,612.5
*exp(0.336*26) = 601,236.6
*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
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 = R² = , n = 44

29. Key Issue: Forecasting Quarterly Temperature in a Resort City for 2005 and 2006
Year Quarter Predicted Temperature
– = ≈ 49
– = ≈ 65
= ≈ 83
≈ 67
– = ≈ 49
– = ≈ 65
= ≈ 83
≈ 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
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
273.5 4
1897
1830
4489 67
310.33 5
1798
1833
1225 35
54.33 6
2168
1847.5
320.5
346.67 7
2364
1983
145161
381
409.67 8
2554
2266
82944
288
444.00 9
3387
2459
861184
928 10
2079
2970.5
891.5
689.33 11
2890
2733
24649
157
216.67 12
2690
2484.5
205.5
95.33
RMSE
MAD
462.0
354.7
490.6
399.04
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 =
- MAD =
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
MAD
AR(2) is the preferred model.