1 ECON203 Principles of Macroeconomics Week 7 Topic: Aggregate Supply Dr. Mazharul Islam.
ECNE610 Managerial Economics Week 4 MARCH 2014 1 Dr. Mazharul Islam Chapter-5.
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Transcript of ECNE610 Managerial Economics Week 4 MARCH 2014 1 Dr. Mazharul Islam Chapter-5.
ECNE610ManagerialEconomics
Week 4
MARCH 2014
1
Chapter-5
Demand Estimation and Forecasting
2
5
Lesson Objectivesknow how to specify and interpret a
regression.understand importance of forecasting
in business.describe six different forecasting
techniques.use seasonal and smoothing methods.recognize limitations of consumer
data.
3
4
Data Sources
PrimaryData Collection
SecondaryData Compilation
Observation
Experimentation
Survey
Print or Electronic
Those that are collected for your purposes.
Data collected & compiled by an outside source or by someone in your organization who provides others access to the data.
TYPES OF DATA
5
Data
Qualitative Quantitative
ContinuousDiscrete
6
Data Timing
Time series data consist of a set of ordered data values observed at successive points in time.
Cross-sectional data are a set of data values observed at a fixed point in time.
7
Data Timing (Panda’s sales reports)
Sales (in $1000’s)
2003 2004 2005 2006
Jeddah 435 460 475 490
Riyadh 320 345 375 395
Dammam
405 390 410 395
Madina 260 270 285 280
Time Series Data
Cross Section Data
8
A Population is the set of all items or individuals of interest. Examples: All likely voters in the next election.
All parts produced today.All sales receipts for November.
A Sample is a subset of the population. Examples: 1000 voters selected at random for
interview.A few parts selected for destructive testingEvery 100th receipt selected for audit.
Populations and Samples
9Introduction to Regression AnalysisRegression analysis is used to:
Predict the value of a dependent variable based on the value of at least one independent variable
Explain the impact of changes in an independent variable on the dependent variable
Dependent variable: the variable we wish to explain
Independent variable: the variable used to explain the dependent variable
15-10
The Multiple Regression Model
Idea: Examine the linear relationship between 1 dependent (y) & 2 or more independent variables (xi)
εxβxβxββy kk22110
kk22110 xbxbxbby
Population model:
Y-intercept Population slopes Random Error
Estimated (or predicted) value of y
Estimated slope coefficients
Estimated multiple regression model:
Estimatedintercept
The Least Squares Equation
The formulas for b1 and b0 are:
21 )x(x
)y)(yx(xb
xbyb 10
and
2i
ii1 )x(x
)y)(yx(xb
2
2
2ˆ
YY
YYR
i
i
ExampleA distributor of frozen desert pies
wants to evaluate factors thought to influence demand.Dependent variable: Pie sales (units per week)Independent variables: Price (in $)
Advertising ($100’s)
Data are collected for 15 weeks
April 21, 2023
Dr. Mazharul Islam
Slide 12
Formulate the ModelWeek
Pie Sales
Price($)
Advertising($100s)
1 350 5.50 3.3
2 460 7.50 3.3
3 350 8.00 3.0
4 430 8.00 4.5
5 350 6.80 3.0
6 380 7.50 4.0
7 430 4.50 3.0
8 470 6.40 3.7
9 450 7.00 3.5
10 490 5.00 4.0
11 340 7.20 3.5
12 300 7.90 3.2
13 440 5.90 4.0
14 450 5.00 3.5
15 300 7.00 2.7
Pie Sales Price Advertising
Pie Sales 1
Price -0.44327 1
Advertising 0.55632 0.03044 1
Correlation matrix:
Estimated Regression model:
April 21, 2023
Dr. Mazharul Islam
Slide 13
Dr. Mazharul Islam15-14
Regression OutputRegression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS FSignificance
F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
CoefficientsStandard
Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales
April 21, 2023
The Regression Equation
ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales where Sales is in number of pies per week Price is in $ Advertising is in $100’s.
April 21, 2023
Dr. Mazharul Islam
Slide 15
Using The Model to Make Predictions
Predict sales for a week in which the selling price is $5.50 and advertising is $350:
Predicted sales is 428.62 pies
428.62
(3.5) 74.131 (5.50) 24.975 - 306.526
ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales
Note that Advertising is in $100’s, so $350 means that x2 = 3.5
April 21, 2023
Dr. Mazharul Islam
Slide 16
15-17
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS FSignificance
F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficient
sStandard
Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
.52148056493.3
29460.0
SST
SSRR 2
52.1% of the variation in pie sales is explained by the variation in price and advertising
Coefficient of Determination(continued)
April 21, 2023 Slide 18
To test the statistical significance of the regression relation between the response variable y and the set of variables x2 and x3, i.e. to choose between the alternatives:
We use the test statistic:0 allnot :
0:0
ia
i
H
H
MSE
MSRF(cal)
Dr. Mazharul Islam
F-Test for Overall Significance of the Model
Shows if there is a linear relationship between all of the x variables considered together and y.
where numerator’s df = k and denominator’s df = (n – k – 1)
)kk,n(αFTabFCal 1;
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
15-19
6.53862252.8
14730.0
MSE
MSRF
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS FSignificance
F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
CoefficientsStandard
Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
(continued)F-Test for Overall Significance
With 2 and 12 degrees of freedom
P-value for the F-Test
5386.62252.8
14730.0
MSE
MSRF
H0: β2 = β3 = 0
HA: β2 and β3 not both zero
= 0.05df2= 2 df3 = 12
Test Statistic:
Decision:
Conclusion:
Reject H0 at = 0.05
The regression model does explain a significant portion of the variation in pie sales
(There is evidence that at least one independent variable affects y )
0
= 0.05
F0.05 = 3.885Reject H0
6.5386MSE
MSRF
Critical Value:
F = 3.885
F-Test for Overall Significance(continued)
FDo not reject H0
April 21, 2023 Slide 21
Significance tests for i
‘t’ test for a population slope• Is there a linear relationship between
x and y ? Null and alternative hypotheses
H0: βi = 0 (no linear relationship)
HA: βi 0 (linear relationship does exist)
Test statistic
iˆse
iβ
iˆ
t
Dr. Mazharul Islam
April 21, 2023 Slide 22
Reject HReject H00 if, if,
Dr. Mazharul Islam
)1;2
( kntcalculatedt
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
15-23
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS FSignificance
F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficient
sStandard
Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
(continued)
Are Individual Variables Significant?
t-value for Price is t = -2.306, with p-value 0.0398
t-value for Advertising is t = 2.855, with p-value 0.0145
d.f. = 15-2-1 = 12
= 0.05
t/2 = 2.1788
Inferences about the Slope: t Test Example
H0: βi = 0
HA: βi 0
The test statistic for each variable falls in the rejection region (p-values < 0.05)
There is evidence that both Price and Advertising affect pie sales at = 0.05
From Excel output:
Reject H0 for each variable
Coefficients Standard Error t Stat P-value
Price -24.97509 10.83213 -2.30565 0.03979
Advertising 74.13096 25.96732 2.85478 0.01449
Decision:
Conclusion:Reject
H0
Reject H0
/2=0.025
-tα/2
Do not reject H0
0tα/2
/2=0.025
-2.1788
2.1788
Forecasting HorizonForecasting Horizon:
The number of future periods covered by a forecast.(Consists of one or more Forecasting Periods.)
It is sometimes referred to as forecast lead-time.
Forecasting Horizon, or lead time, is typically divided into four categories.
Immediate term – less than one month Short term – one to three months Medium term – three months to two years Long term – two years or more
Time-Series Components
Time-Series
Cyclical Component
Random Component
Trend Component
Seasonal Component
Upward trend
Trend Component Long-run increase or decrease over time
(overall upward or downward movement) Data taken over a long period of time
Sales
Time
Downward linear trend
Trend Component Trend can be upward or downward Trend can be linear or non-linear Can be stationary or non-stationary
Sales
Time Upward nonlinear trend
Sales
Time
(continued)
Seasonal Component Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Recurrence period - shortest period of repetition
(must be less than one year)
Sales
Time (Quarterly)
Winter
Spring
Summer
Fall
Cyclical Component Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to
trough
Sales
1 Cycle
Year
Random Component Unpredictable, random, “residual” fluctuations Due to random variations of
Nature Accidents or unusual events
“Noise” in the time series
Multiplicative Time-Series Model
Used primarily for forecasting Allows consideration of seasonal
variation Observed value in time series is the
product of components Classical decomposition is used to
identify the various components
where Tt = Trend value at time t
St = Seasonal value at time t
Ct = Cyclical value at time t
It = Irregular (random) value at time t
ttttt ICSTy
16-33
Seasonal Adjustment
1. Compute each moving average2. Compute the centered moving averages3. Isolate the seasonal component by determining
the ratio-to-moving average values4. Determine seasonal indexes and normalize if
necessary5. Deseasonalize the time series6. Develop trend line using deseasonalized data7. Develop unadjusted forecasts using trend
projection8. Seasonally adjust the forecasts
16-34
Moving AveragesExample: Four-quarter moving average
First average:
Second average:
etc…
(continued)
4
Q4Q3Q2Q1average Moving 1
4
Q5Q4Q3Q2average Moving 2
16-35
Seasonal Data
Quarter Sales
1
2
3
4
5
6
7
8
9
10
11
etc…
23
40
25
27
32
48
33
37
37
50
40
etc…
Quarterly Sales
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Quarter
Sal
es…
…
16-36
Calculating Moving Averages
Each moving average is for a consecutive block of 4 quarters
Quarter Sales
1 23
2 40
3 25
4 27
5 32
6 48
7 33
8 37
9 37
10 50
11 40
Average Period
4-Quarter Moving
Average
2.5 28.75
3.5 31.00
4.5 33.00
5.5 35.00
6.5 37.50
7.5 38.75
8.5 39.25
9.5 41.00
4
43212.5
4
2725402328.75
etc…
16-37
Centered Moving Averages
Average periods of 2.5 or 3.5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages
Average Period
4-Quarter Moving
Average
2.5 28.75
3.5 31.00
4.5 33.00
5.5 35.00
6.5 37.50
7.5 38.75
8.5 39.25
9.5 41.00
Centered Period
Centered Moving
Average
3 29.88
4 32.00
5 34.00
6 36.25
7 38.13
8 39.00
9 40.13
etc…
16-38
Calculating the Ratio-to-Moving Average
Divide the actual sales value by the centered moving average for that quarter
16-39
Calculating Seasonal Indexes
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
0.837
0.844
0.941
1.324
0.865
0.949
0.922
etc…
…
…
88.29
25837.0
Example:
16-40
Calculating Seasonal Indexes
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
0.837
0.844
0.941
1.324
0.865
0.949
0.922
etc…
…
…
Average all of the Fall values to get Fall’s seasonal index
Fall
Fall
Fall
Do the same for the other three seasons to get the other seasonal indexes
(continued)
16-41
Interpreting Seasonal Indexes
Suppose we get these seasonal indexes:
SeasonSeasonal
Index
Spring 0.825
Summer 1.310
Fall 0.920
Winter 0.945
= 4.000 -- four seasons, so must sum to 4
Spring sales average 82.5% of the annual average sales
Summer sales are 31.0% higher than the annual average sales
etc…
Interpretation:
Since the sum 4 use a multiplier
averages of Sum
4 Multiplier
To find the Seasonal Index
Adjusted seasonal indexes are:
These seasonal indexes can now be used to remove the seasonal component from the original series
Winter Spring Summer Fall1.4014 0.6078 1.2732 0.65121.4468 0.5836 1.3592 0.64471.4754 0.6318 1.3368 0.5809
Average 1.4412 0.6077 1.3231 0.6256 3.9976 SumAdjusted SI 1.4420 0.6081 1.3239 0.6260 4.0000 Sum
16-44
Deseasonalizing
The data is deseasonalized by dividing the observed value by its seasonal index
This smooths the data by removing seasonal variation
16-45
DeseasonalizingQuarter Sales
Seasonal Index
Deseasonalized Sales
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
0.825
1.310
0.920
0.945
0.825
1.310
0.920
0.945
0.825
1.310
0.920
…
27.88
30.53
27.17
28.57
38.79
36.64
35.87
39.15
44.85
38.17
43.48
…
0.825
2327.88
etc…
(continued)
Example:
16-46
Unseasonalized vs. Seasonalized
Sales: Unseasonalized vs. Seasonalized
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11Quarter
Sal
es
Sales Deseasonalized Sales
Fitting trend models
Once the seasonality has been removed from the data set, trend models can be applied to the deseasonalised data
Trend-Based Forecasting Estimate a trend line using regression analysis
Year
Time Period
(t)Sales
(y)
1999
2000
2001
2002
2003
2004
1
2
3
4
5
6
20
40
30
50
70
65
tbby 10
Use time (t) as the independent variable:
Trend-Based Forecasting The linear trend model is:
Sales trend
01020304050607080
0 1 2 3 4 5 6 7
Year
sale
s
Year
Time Period
(t)Sales
(y)
1999
2000
2001
2002
2003
2004
1
2
3
4
5
6
20
40
30
50
70
65
years in measured ist and 1999 1, t where
t 9.571412.333Sales
(continued)
Trend-Based Forecasting Forecast for 2005 (ie t = 7):
Sales
01020304050607080
0 1 2 3 4 5 6 7
Year
sale
s
Year
Time Period
(t)Sales
(y)
1999
2000
2001
2002
2003
2004
2005
1
2
3
4
5
6
7
20
40
30
50
70
65
??
(continued)
33.79
(7) 5714.9333.12Sales