# Forecasting

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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)
* 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 expt             ln( yt )   0   1 t  t where  0  ln( )
Note : exp1t  e1t and expt  et

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

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

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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