二元线性回归模型案例

1、二元线性回归模型范例我国货物周转量问题研究。一、建立模型通过经济分析可知，工农业总产值、运输线路长度是影响我国货物周转量的主要因素。用y表示货物周转量， x1表示工农业总产值，x2表示运输线路长度，可建立如下二元线性回归理论模型yi = 3 + Pxii +02X21 +叫（i）收集我国某一时间13年的货物运输量（单位：1亿吨公里）、工农业总产值（单位：亿 元）、运输线路长度（单位：万公里）的统计资料数据，并同时给出离差形式数据。年份货物周 转量V国内生产总值X1i运输线 路长度X2i一X1i = X1i X1X2i=Xz -X2一yi = yi - y12236150469.54-4328.

2、8461-42.7446-6726.538423463223574.79-3597.8461-37.4946-5499.538434565313886.79-2694.8461-25.4946-4397.5384472974467105.47-1365.8461-6.8146-1665.5384569044536111.12-1296.8461-1.1646-2058.5384679694978117.92-854.84615.6354-993.5384768295634123.2-198.846110.9154866.46168109076379120.25546.15397.9654194

3、4.46169115177077124.711244.153912.42542554.461610116167580128.431747.153916.14542653.461611124038291130.922458.153918.63543440.461612132959211131.563378.153919.27544332.46161314512107971354964.153922.71545549.4616工工 ny =8962.5384X1 =5832.8461X2 =112.2846二、参数估计多元线性回归模型的参数估计公式:其中所以死二二二上 nk-1 nk1Cov(M)

4、 =c-i(X X),、ei2Cii n -k -1 ci n - k -1其中，。为（XX ）,主对角线上元素。s"v"：)57yzi(3)(4)(5)(6)对于二元线性回归模型，由正规方程组可以得到参数估计公式如下2(三Xii yi)(三X2i) -(三X2i yi)(三Xii xQ2 J (三 Xli)(三 X2i) (三 Xii X2i)2 / 2i yi)(三X1i) -(1X1 yi)(ZX1i X2i)J J.(三 X1i)(三 X2i) (，X1iX2i)2A -1 Xi -X2由于(X X )方 2- X2iL- X1i X2iL- X1i X2iL 2&

5、#39;、%L(7)(8) 2L =(" X1i )C2 2,X2i) 一(X1i X2i )2(91)三X2iG1二.2.(三X1i)(三X2i) -(。X2i)2三XliC22 =.2.2.（三 X1i）（三 X2i） 一（三 XiiX）统计计算表如下:年 份2X1i2X2i2 V X1i y, X2i y X1i X2i118738908.561827.100845246318.85299118149.52287523.1933185034.7950212944496.561405.845030244922.611978.6492.78206202.9925134899.800

6、437262195.503649.974619338343.9811850689.21112113.482568704.023441865535.56946.43882774018.1622274869.12811349.97809307.694851681809.8071.35634237580.3442669607.4962397.37381510.30706730761.854731.7577987118.552389323.4264-5598.9863-4817.3997739539.7715119.1460750755.7043-172292.59457.7750-2170.4847

7、8298284.082563.44763780930.9141061975.28515488.41444350.334391547918.927154.39066525274.0563178143.36231740.207215459.1099103052546.75260.67407040858.4634636005.78342841.198928208.4986116042520.596347.278111836776.028457184.164114.378145808.68121211411923.94371.541018770223.5214635722.0583509.930365

8、115.26771324642823.94515.989430796524.0527548381.44126058.2400112762.7415工90259265.695794.9399（TSS）182329645.2125894250.1987198.1777664173.3692计算各参数：由公式（7）得到0.9018=66.9937125894250.1 5794.9399 -987198.1777 664173.3692'1 =I -290259265.69 5794.9399 -664173.3692987198.1777 90259265.69 -125894250.1

9、664173.3692Z_ "" ""_290259265.69 5794.9399 -664173.3692% =8962.5384-0.9018 5832.8461-66.9939 112.2846= -3820.078根据上述计算结果， 二元回归模型 为:yi =-3820.078 0.9018X1i 66.9937X2i残差计算如下年 份A yiA.3 = yi - yi2ei12194.971141.02891683.370823205.9038257.096266098.444234824.1536-259.153667160.6003472

10、74.088122.9119524.953457714.8267-810.8267657440.008868568.9795-599.9795359975.405279514.2870314.713099044.247289988.4966918.5034843648.4499910916.7450600.25513603065669-3.566912.72271112427.561-24.5610603.24291751213300.0930-5.093025.93841314960.8061-448.8061201426.9154y(RSS) 2657950.45

11、2由公式(3) ( 6),得到2 一2657950.452c .1 二10265795.0452Var( -1) =5794.93992 265795.045290259265.69 5794.9399 -664173.3692=0.0188019Var( '-2)=90259265.6990259265.69 5794.9399 -664173.36922265795.0452=292.8496617S - =0.13712004S . =17.112851三、拟合优度的检验tss=x (yi -y)2 2=' yi =182329661.6RSS=' (y -yi)

12、2=" 0 2=2657950.452、.， 二 2ESS (yi -y)=TSS-RSS=182329645.2 - 2657950.452=179671716.248-1Sr =RSS =2657950.452 - 10 =265795.0452n - k -11St = TSS =182329645.2 -12=15194137.1n -12SrR2 =1 - =1St265795.0452=0.98250715194137.1四、显著性检验1. F检验337.98921179671716.248/22657950.452/10若给定 a =0.05,查 F (2, 10)的临

13、界值，F0.05 =4.10。显然F=337.98920 > F0.05 =4.10,因此回归方程显著成立tit1t2SiciiC11£ e2n -k -1?10.9018，ei20.137120046.577010?266.9937C22、:e2 17.1128513.914810对给定的口 =0.05,查 t0.05(10)的临界值，t°.025(10)= 2.228。由于tt2均大于t0.025 (10),因此两个参数百、P2显著的不为零Eviews软件计算结果Dependent Variable: 丫Method: Least SquaresDate: 12/08/03 Time: 19:05Sample: 2001 2013Included observations: 13VariableCoefficientStd. Errort-Statistic Prob.X10.9018330.1371206.5769640.0001X266.9937017.112853.9148180.0029C-3820.0781236.779-3.0887310.0115R-squared0.985422Mean dependent var8962.538Adjusted R-squared0.982507S.D.