The Art of Mathematics in Business. Dr Jae K Shim
rel="nofollow" href="#fb3_img_img_d25cf8ee-84f8-58c8-8279-bfb74e215d07.jpg" alt="image"/>
(1)R-squared (r2) = .608373 = 60.84%
(2)Standard error of the estimate (Se) = 2.343622
(3)Standard error of the coefficient (Sb) = 0.142893
(4)t-value = 3.94
Note that all of the above are the same as the ones manually obtained. Note the following:
(1)t-statistic is more relevant to multiple regressions which have more than one b’s.
(2)r2 tells you how good the forest (overall fit) is while t-statistic tells you how good an individual tree (an independent variable) is.
In summary, the table t value, based on a degree of freedom and a level of significance, is used:
1.To set the prediction range—upper and lower limits—for the predicted value of the dependent variable.
2.To set the confidence range for regression coefficients.
3.As a cutoff value for the t-test.
21. Simple Regression
Introduction
Simple regression is a type of regression analysis that involves one independent (explanatory) variable.
How is it computed?
Simple regression takes the form:
Y = a + bX
Where Y = dependent variable
X = independent variable
a = constant
b = slope
The least-squares estimation method is typically used to estimate the parameter values a and b.
Example
Assume that data on DVD sales and advertising expenditures have been collected over the past seven periods. The linear regression equation can be estimated using the least-squares method. For example, the sales/advertising regression for DVDs is estimated to be:
DVD sales = Y = 19.88 + 4.72X
r2 = 0.7630 = 76.30%
How is it used and applied?
Other applications of simple regression are:
• Total manufacturing costs is explained by only one activity variable (such as either production volume or machine hours), i.e., TC = a + b Q.
• A security’s return is a function of the return on a market portfolio (such as Standard & Poor’s 500), i.e., rj = a +βrm where β = beta, a measure of uncontrollable risk.
• Consumption is a function of disposable income, i.e., C = a + b Yd where b = marginal propensity to consume.
• Demand is a function of price, i.e., D = a - bP.
• Average time to be taken is a function of cumulative production, i.e., Y = a X-bwhere b represents a learning rate in the learning curve phenomenon.
• Trend analysis that attempts to detect a growing or declining trend of time series data, i.e., Y = a + bt where t = time.
22. Trend Equation
Introduction
Trends are the general upward or downward movements of the average over time. These movements may require many years of data to determine or describe them. The basic forces underlying trends include technological advances, productivity changes, inflation, and population change.
How is it computed?
The trend equation is a common method for forecasting sales or earnings. It involves a regression whereby a trend line is fitted to a time series of data.
The linear trend line equation can be shown as
Y = a + bX
The formulas for the coefficients a and b essentially the same as for simple regression. They are estimated using the least-squares method, which was discussed in Sec. 19, Regression Analysis.
However, for regression purposes, a time period can be given a number so that ΣX = 0. When there is an odd number of periods, the period in the middle is assigned a value of 0. If there is an even number, then − 1 and + 1 are assigned the two periods in the middle, so that again ∑X = 0.
With ∑X = 0, the formula for b and a reduces to
Example 1
This example demonstrates the use of trend equations in cases which an even number and an odd number of periods occur.
Case 1 (odd number):
Case 1 (odd number):
Example 2
| Year | Sales (in millions) |
| 20×1 | $ 10 |
| 20×2 | 12 |
| 20×3 | 13 |
| 20×4 | 16 |
| 20×5 | 17 |
Since the company has five years of data, which is an odd number, the year in the middle is assigned a zero value.
Therefore, the estimated trend equation is
Y′ = $13.6 + $1.8 t
To project 20×6 sales, we assign +3 to the t value for the year 20×6.
Y’ = $13.6 + $1.8 (3)
= $19
How is it used and applied?
Managers use the trend equation for forecasting purposes, such as to project future revenue costs. They should use the trend equation, however, only if the time series data reflect the gradual shifting or growth pattern over time.
See also Sec. 23, Decomposition of Time Series.
23. Decomposition of Time Series
Introduction
When sales exhibit seasonal or cyclical fluctuation, we use a forecasting method called classical decomposition to deal with seasonal trend, and cyclical components together.
How is it computed?
We assume that a time series (Yt) is combined into a model that consists of the four components--trend (T), cyclical, (C), seasonal (S), and random (R). This model is of a multiplicative type i.e.,
Yt = T × C × S × R
The classical decomposition method is illustrated step by step, by working with the quarterly sales data. The approach basically requires the following four steps:
1.Determine seasonal indexes, using a four-quarter moving average.
2.Deseasonalize the data.
3.Develop the linear least-squares equation in order to identify the trend component of the forecast.
4.Forecast the sales for each of the four quarters of the coming year.
Example
We illustrate the classical decomposition approach by working with the quarterly sales data presented in