4. Mathematical Representations

The objective of mathematical representations is to determine relationship among variables and develop mathematical model that reflects how the firm demand is affected by different variables.

 Two analysis methods can be employed to discover the relationship – time series and regression analysis. Time series analysis is a simple measure of how dependent variable changes against the lapse of time. Time will be the only independent variable in the analysis. Regression analysis attempts to depict the relationships between independent and dependent variables so that dependent variable can be predicted. The relationship can be linear, exponential, power, polynomial, etc. There can one independent variable, as well as multiple variables.

In this project, our objective is to determine how exogenous, endogenous, and firm specific variables combined affect the firm demand. 

1)      Firm Demand (FD). The firm demand is calculated by the product of the total industry demand (TID) and the firm market share (MS).

= TID * MS

2)      Total Industry Demand (TID). Total industry demand is expressed by the average firm demand (AFD) and the number of firms in the industry (N).

= AFD * N

Combine this formula with the formula in 1), the firm demand can be expressed

= AFD * N * MS

To simplify the formula, we create a new variable, called Normalized Market Share (NSOM) derived by N * MS. The final firm demand formula looks like this

=AFD * NSOM

3)      Average Firm Demand (AFD). To explore the use of regression modeling for obtaining more reliable forecasts, scatter plot is a simple start point to visualize if there are any patterns in the industry demand. 

Figure 1 – Average Firm Demand Time Series 

From the above scatter spot chart, we clearly see a pattern that the average firm demand is in an upper-tick trend as time is passing by. Time series analysis will be a perfect fit for forecasting the future demand. By extending the trend we will have a fairly good estimate of AFD. 

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

Multiple R

0.685774645

 

 

 

 

 

R Square

0.470286863

 

 

 

 

 

Adjusted R Square

0.443801206

 

 

 

 

 

Standard Error

443.2454672

 

 

 

 

 

Observations

22

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

Regression

1

3488515.89

3488515.89

17.75628469

0.000426705

 

Residual

20

3929330.883

196466.5442

 

 

 

Total

21

7417846.773

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

1366.87013

195.6341059

6.986870328

8.83431E-07

958.7847254

1774.955534

PRD

62.76623377

14.89532626

4.213820676

0.000426705

31.69514209

93.83732545

Table 1 – Average Firm Demand Time Series Result 

AFD = 62.766 * T + 1366.9 (equation 1) 

We could use Crystal Ball Predictor 1.3 to perform time series analysis as well. Predictor actually provides more accurate time forecast compared with linear regression analysis because it can follow the seasonality embedded in the original data while linear regression analysis tries to represent the trend by one straight line. The detailed comparison between the two can be found in the file CB_Forecast_Avg_Dem.xls. For the simplicity of our model, we adopted the result from the linear regression analysis, which still gives us a fairly good estimate of the relationship between the demand and time. 

However, the time is not the only factor explaining the changes in the demand. This is testified by the fairly large residuals between historical data and predicted values by the analysis (See demand.xls). We have to introduce other factors, such as Average Price (Avg_Price), Average Advertising Expenditures (Avg_Adv), and Average R&D (Avg_RD), to analyze their influences on the residuals. Both Excel Regression Analysis function and SAS Stepwise procedure can help us find the relationship.

a.       Regression Analysis by Using Excel

Using Excel to eliminate variables that are not significantly related to the dependent variable can be extremely cumbersome if lots of independent variables are involved to be tested. We have to run regression analysis several times to obtain the final formula, each time eliminating one variable with p-value greater than alpha. In our case, we didn’t get the ideal result until regression analysis was run the 6th time. 

Residuals = -(43.8 * Avg_Price – 16429) (equation 2) 

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

Multiple R

0.533422954

 

 

 

 

 

R Square

0.284540048

 

 

 

 

 

Adjusted R Square

0.248767051

 

 

 

 

 

Standard Error

374.9185834

 

 

 

 

 

Observations

22

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

Regression

1

1118051.999

1118051.999

7.954045436

0.010571406

 

Residual

20

2811278.884

140563.9442

 

 

 

Total

21

3929330.883

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-16429.1921

5825.910028

-2.820021598

0.010577736

-28581.82182

-4276.562385

Avg_Price

43.80010231

15.53034456

2.820291729

0.010571406

11.40438628

76.19581834

Table 2 – Average Firm Demand Residual Regression Result


                                Figure 2 – Average Firm Demand Residual Regression 

b. Regression Analysis by Using SAS

SAS provides built-in procedure – stepwise that will give the test results within seconds. When analyzing relationship by using stepwise procedure, we want to use both forward and backward tests. Ideally, two tests should result in the same model. If two different models are produced after two tests, we need to go back and check our data, modeling process, and assumptions. If no mistake is found, we have to use our best judgment to determine which one will be the best-fitted model.

Fortunately, two tests result in the same regression formulas for our model. The only variable left in the model that is significant at the 0.05 level is Avg_Price. 

So far we have determined the mathematical relationship among variables. Combine equation 1 and 2 together, we get the Average Firm Demand in term of time and Average Price. 

AFD = 62.766 * T - 43.8 * Avg_Price + 17796.09 (equation 3) 

4)      Normalized Market Share (NSOM). The more challenging variable for prediction is the firm’s Market Share (MS). This variable is dependent on the firm’s ability to compete effectively. Competition is fairly intense and is based on pricing, promotion, and loyalty. NSOM is the relative demand to the industry average, therefore the predictor variables should also be relative to the average for the industry. Hence, we have computed Prel (relative price) and Arel (relative advertising). The previous quarter’s relative demand (NSOM1) is used as a measure of Brand Loyalty. To measure the lagging effects on the demand of previous advertising expenditures and R&D expenditures, the last quarter’s Arel1 and Rrel1 and Arel2 and Rrel2 before last quarter are included in the analysis. 

Prel  = Firm Price / Average Price

Arel = Firm Advertising Expenditures / Average Advertising Expenditures

NSOM1 = Last Quarter’s relative Demand 

The result from the regression analysis is listed in the following table. As the p-values of all the variables are far less than the significant level 0.05, all variables are left in the model for predication. 

NSOM = 15.95 – 16.79 * Prel + 0.73 * Arel + 0.44 * NSOM1 + 0.18 * Rrel1 + 0.26 * Arel1 + 0.15 * Rrel2 + 0.08 * Arel2 (equation 4) 

 SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

Multiple R

0.993207601

 

 

 

 

 

R Square

0.986461339

 

 

 

 

 

Adjusted R Square

0.985910347

 

 

 

 

 

Standard Error

0.03190622

 

 

 

 

 

Observations

180

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

Regression

7

12.75802658

1.822575225

1790.33686

4.339E-157

 

Residual

172

0.175097182

0.001018007

 

 

 

Total

179

12.93312376

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

15.94723159

0.259761394

61.39184634

6.7113E-119

15.43450063

16.45996256

Prel

-16.79336984

0.255932657

-65.61636184

1.1436E-123

-17.29854344

-16.28819624

Arel

0.7312239

0.015568958

46.9667847

5.0441E-100

0.700493056

0.761954744

NSOM1

0.444773948

0.011004677

40.41680965

9.83333E-90

0.423052326

0.466495571

Rrel1

0.178938881

0.026025688

6.875471742

1.09384E-10

0.127567983

0.230309779

Arel1

0.256960344

0.017241801

14.90333562

8.68077E-33

0.222927553

0.290993135

Rrel2

0.152087781

0.027751864

5.480272633

1.49013E-07

0.097309664

0.206865898

Arel2

0.079412054

0.017024876

4.664471878

6.19055E-06

0.045807442

0.113016667

Table 3 – Normalized Market Share Regression Result  

The model developed above is quite complicated. From the result table, we can see that NSOM1, Rrel1, Arel1, Rrel2, and Arel2 do not have strong correlations with the NSOM because their coefficients are relative small compared with those of Prel and Arel. By instinct, those variables may be eliminated without detriment to the significance of the model. We prove our thought by running the regression analysis again and get the following result.

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

Multiple R

0.837757798

 

 

 

 

 

R Square

0.701838128

 

 

 

 

 

Adjusted R Square

0.698469067

 

 

 

 

 

Standard Error

0.147601624

 

 

 

 

 

Observations

180

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

Regression

2

9.076959369

4.538479684

208.3186355

3.08327E-47

 

Residual

177

3.85616439

0.021786239

 

 

 

Total

179

12.93312376

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

20.56532439

1.11042456

18.52023553

2.79161E-43

18.37394818

22.75670061

Prel

-20.4323178

1.120634508

-18.23281155

1.72581E-42

-22.64384292

-18.22079269

Arel

0.865031229

0.070661781

12.24185428

2.3341E-25

0.725583173

1.004479286

Table 4 - Normalized Market Share Regression Result 1

The significance F of the regression is 3.08327E-47 and Adjusted R Square is 0.69469067, which justifies our thought that the model is still a good fit with only Prel and Arel left in the model. Therefore, our final formula for the NSOM is

NSOM = 20.565 – 20.432 * Prel + 0.865 * Arel (equation 5) 


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