店铺租金的确定模型
某商人欲在某火车站附近经营一店铺，委托本小组对相关情况进行调查。经过数月的资料收集和整理，我们的调查成果如下：
进出车站的乘客为主要服务对象的10家便利店的数据。
Y是日均销售额，X1为店铺面积，X2是店铺距车站的距离，X3为店员人数，X4为店铺日租金。
具体数据如下表：
店铺代码 日均销售额（元）Y 店铺面积（m2）X1 离车站距离（100m）X2 店员人数（人）X3 店铺日租金（元）X4
A
B
C
D
E
F
G
H
I
J 4000
4500
8000
6000
5000
2000
1500
9000
3000
7000 60
100
85
50
75
55
70
95
45
65 3
5
2
1
3
4
6
1
3
2 5
7
5
3
5
4
5
6
4
4 600
600
1020
750
750
440
280
1425
450
780
数据来源：www.sina.com.cn
为了考察店铺面积、离车站距离、店员人数和日租金对日销售额的影响，我们首先做Y关于X1、X2、X3、X4的回归，即建立如下回归模型：
Y=C+β1 X1+β2 X2+β3 X3+β4 X4
得回归结果如下表：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 17:51
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 4815.267 1536.418 3.134087 0.0258
X1 128.1930 39.79796 3.221096 0.0234
X2 -1494.966 513.4078 -2.911848 0.0333
X3 -619.1674 472.6664 -1.309946 0.2472
X4 -1.877208 2.938471 -0.638838 0.5510
R-squared 0.970270 Mean dependent var 5000.000
Adjusted R-squared 0.946486 S.D. dependent var 2505.549
S.E. of regression 579.6124 Akaike info criterion 15.86945
Sum squared resid 1679752. Schwarz criterion 16.02074
Log likelihood -74.34724 F-statistic 40.79489
Durbin-Watson stat 1.407218 Prob(F-statistic) 0.000522
从回归结果来看， R2接近于1，整个方程的拟合优度很高，F>F0.05(4，5)＝5.19，变量X3、X4对应的偏回归系数之t值小于2，而且X3、X4的符号与经济意义相悖，该模型明显存在多重共线性，回归结果不显著，回归方程不能投入使用。
由于变量较多，采用逐步回归法来修正模型。
用Y对各个变量单独进行回归：
对X1，有：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 20:17
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 444.4444 2988.555 0.148716 0.8855
X1 65.07937 41.38415 1.572567 0.1545
R-squared 0.236129 Mean dependent var 5000.000
Adjusted R-squared 0.140645 S.D. dependent var 2505.549
S.E. of regression 2322.680 Akaike info criterion 18.51569
Sum squared resid 43158730 Schwarz criterion 18.57620
Log likelihood -90.57844 F-statistic 2.472968
Durbin-Watson stat 1.988381 Prob(F-statistic) 0.154464
对X2，有：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 20:20
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 8687.500 1096.232 7.924871 0.0000
X2 -1229.167 324.6760 -3.785826 0.0053
R-squared 0.641777 Mean dependent var 5000.000
Adjusted R-squared 0.596999 S.D. dependent var 2505.549
S.E. of regression 1590.581 Akaike info criterion 17.75844
Sum squared resid 20239583 Schwarz criterion 17.81896
Log likelihood -86.79221 F-statistic 14.33248
Durbin-Watson stat 2.488527 Prob(F-statistic) 0.005344
对X3，有：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 20:28
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 3344.828 3791.325 0.882232 0.4034
X3 344.8276 770.6964 0.447423 0.6664
R-squared 0.024413 Mean dependent var 5000.000
Adjusted R-squared -0.097536 S.D. dependent var 2505.549
S.E. of regression 2624.897 Akaike info criterion 18.76033
Sum squared resid 55120690 Schwarz criterion 18.82084
Log likelihood -91.80164 F-statistic 0.200188
Durbin-Watson stat 2.273575 Prob(F-statistic) 0.666436
对X4，有：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 20:30
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C -124.4556 691.7552 -0.179913 0.8617
X4 7.222630 0.893132 8.086854 0.0000
R-squared 0.891004 Mean dependent var 5000.000
Adjusted R-squared 0.877380 S.D. dependent var 2505.549
S.E. of regression 877.3734 Akaike info criterion 16.56860
Sum squared resid 6158272. Schwarz criterion 16.62912
Log likelihood -80.84299 F-statistic 65.39721
Durbin-Watson stat 1.099477 Prob(F-statistic) 0.000040
从上面的回归结果可以看到，Y对X2的回归拟合最好，故选择该回归式为基本回归表达式。现在分别加入X1、X3、X4回归结果如下：
加入X1，有：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 21:21
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 3641.214 817.1938 4.455753 0.0030
X1 75.45849 10.58869 7.126326 0.0002
X2 -1307.769 121.3087 -10.78050 0.0000
R-squared 0.956605 Mean dependent var 5000.000
Adjusted R-squared 0.944206 S.D. dependent var 2505.549
S.E. of regression 591.8273 Akaike info criterion 15.84763
Sum squared resid 2451817. Schwarz criterion 15.93841
Log likelihood -76.23816 F-statistic 77.15446
Durbin-Watson stat 1.809788 Prob(F-statistic) 0.000017
可见，加入X1效果较好，这样回归式中就有X1、X2两个变量了。在此基础上继续加入其他变量。
加入X3，有：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 21:26
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 3993.580 797.8410 5.005484 0.0024
X1 109.3747 25.40691 4.304920 0.0051
X2 -1181.338 142.6370 -8.282130 0.0002
X3 -647.0407 446.8316 -1.448064 0.1978
R-squared 0.967843 Mean dependent var 5000.000
Adjusted R-squared 0.951765 S.D. dependent var 2505.549
S.E. of regression 550.2815 Akaike info criterion 15.74791
Sum squared resid 1816859. Schwarz criterion 15.86895
Log likelihood -74.73956 F-statistic 60.19526
Durbin-Watson stat 1.281362 Prob(F-statistic) 0.000072
可以看出，加入了X3以后引起了多重共线性，故剔除。
现在加入X4，回归结果如下：
Dependent Variable: Y
Method: Least Squares
Date: 12/14/03 Time: 21:29
Sample: 1 10
Included observations: 10
Variable Coefficient Std. Error t-Statistic Prob.
C 4636.482 1619.077 2.863658 0.0287
X1 99.57632 35.19507 2.829269 0.0300
X2 -1674.283 523.5131 -3.198167 0.0186
X4 -2.232526 3.095576 -0.721199 0.4979
R-squared 0.960067 Mean dependent var 5000.000
Adjusted R-squared 0.940100 S.D. dependent var 2505.549
S.E. of regression 613.2195 Akaike info criterion 15.96450
Sum squared resid 2256229. Schwarz criterion 16.08553
Log likelihood -75.82249 F-statistic 48.08356
Durbin-Watson stat 1.907328 Prob(F-statistic) 0.000137
同样，X4引起多重共线性，故剔除。
故Y对X1、X2的回归拟合最好，回归表达式应为：
Y=3641.214+75.45849X1－1307.769X2
其经济意义为，在其他条件不变时，店铺面积扩大1平方米，日均销售额大约会增加75.5元；店铺如果比现在地址再远离车站100米，日均销售额大约会减少1307.8元。
由于客户的资金有限，每天能负担的租金为700～800元，因此我们建议在离火车站100米处租赁面积为60平方米左右的店铺，租金大约为750元。这样客户能够获得既定条件下的最大收益。
以上就是我们的分析报告。