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Keating_Chapter_4_Extrapolation_2.pptxChapter 4 Extrapolation 2: Introduction to Forecasting with Regression Trend Models
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A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x).
Can be used to answer questions such as...Does y tend to increase when x increases?
Describes the way in which one variable is related to another (e.g. Sales and Price).
Regression Models (Demand Planning Models)
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Many sophisticated software packages are available -- but there is a danger in using canned packages unless you are familiar with the underlying concepts.
Today’s software packages are easy to employ, but you need to learn the underlying concepts.
Regression Models (Demand Planning Models)
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A regression model is a simplified or ideal representation of the real world.
All scientific inquiry is based to some extent on models - that is the set of simplifying assumptions - on which regression is based.
Regression Models (Demand Planning Models)
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Y = a + b X
Y = Dependent Variable
X = Independent Variable
a = Intercept of the line
b = Slope of the line
Least Squares Linear Regression
When the term “Regression” is used, this is the model that is being referenced.
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Figure 4.1
It is important to look at the data before plunging into data analysis and the selection of an appropriate set of forecasting techniques.
Figure 4.3 A Trend Regression
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Audit Report result from ForecastX
In most cases it is best to use the Audit Report to examine results in order to see all the diagnostic statistics in one location.
Data for a Causal Regression
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Figure 4.5 Jewelry Sales Causal Regression
Data for 2010–2015 were used to estimate this model. The positive slope (0.91) indicates that, on average, JS increases by $0.91 million for each additional $1 billion increase in disposable personal income.
A major problem with this model is apparent in the graph. It is clear from the graph of actual and predicted retail sales that this model fails to deal with the seasonality in JS.
Figures 4.7 and 4.8 Seasonally Adjusted Jewelry Sales Model and Reseasonalized Forecast Values
Four Quick Checks (for regression models)
If you perform 4 quick checks you will rarely misinterpret the results of a regression estimate.
The checks require no reference tables or calculations.
Failure to perform the checks could lead to consistent misinterpretation of the results.
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We will add a fifth quick check in the following chapter.
1) Does the model make economic sense (i.e., check slope term)?
2) Is there a statistically significant relationship between the dependent and independent variables (t-test)?
3) What percentage of the variation in the dependent variable does the regression model explain (R-Square)?
4) Is there a problem of serial correlation among the error terms in the model (Durbin-Watson)?
Four Quick Checks
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1) check slope term
2) t-test
3) R-Square
4) Durbin-Watson
Four Quick Checks
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Economic Meaning
The Slope Term (what is its sign?)
There is almost always an economic or business logic that indicates whether the relationship between the dependent variable (Y) and an independent variable (X) should be positive or negative.
Does the model make economic sense?
First Quick Check
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Use a t-test to examine the null hypothesis that the slope of the true relationship between X and Y is equal to zero.
Hint: if the t-statistic is greater than 2, the coefficient is statistically significant (rule of thumb).
Is the coefficient statistically significant?
Second Quick Check
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Third Quick Check
Coefficient of Determination (“R-Squared”)
The third check of regression results is to evaluate what percent of the variation (i.e., up-and-down movement) in the dependent variable is explained by variation in the independent variable.
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Business and economic data is most often time series data. In using regression analysis with time series data a problem known as autocorrelation can cause some difficulty.
Autocorrelation (serial correlation) is a significant pattern in the error terms or residuals.
If present, the t-ratios will be overstated and the F-statistic and R-Squared statistics will be unreliable.
A Fourth Quick Check (perhaps the most important check?)
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Check for autocorrelation with the Durbin-Watson.
Fourth Quick Check
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Rule of Thumb:
If DW(1) is between 1.5 and 2.5, you probably
have no 1st order serial correlation (autocorrelation).
Fourth Quick Check
Durbin Watson Statistic
The DW statistic will always be in the range of 0 to 4. As a rule of thumb, a value close to 2 (e.g., between 1.50 and 2.50) indicates that there is no serial correlation. When the DW statistic approaches 4, the degree of negative serial correlation increases. When positive serial correlation exists, the value of DW approaches 0.
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Figure 4.9 The Standard Error of the Estimate
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The standard error of the estimate (SEE) can be used to generate approximate confidence intervals with relative ease.
Figure 4.13 Heteroscedasticity
One of the assumptions of regression analysis is that the error terms in the population regression (εi) have a constant variance across all values of the independent variable (X). When this is true, the model is said to be homoscedastic,and if this assumption is violated, the model is termed heteroscedastic.
Figure 4.13 Cross- Sectional Regression
While most forecasting is based on time-series data, there are situations in which cross-sectional analysis is useful. In cross-sectional analysis, the data all pertain to one time period rather than a sequence of periods.
Cross- Sectional Regression
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While most forecasting is based on time-series data, there are situations in which cross-sectional analysis is useful. In cross-sectional analysis, the data all pertain to one time period rather than a sequence of periods.
Figure 4.15
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Table 4.15
Trend Forecast
Figure 4.16
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Table 4.16
Causal Forecast
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Intergrative Case: The Gap
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Trend Forecast: The Gap
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Linear Trend with Seasonality
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Total Variation in Y
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