在回归分析中, 解释变量的个数应当是越少越好, 因为模型越简洁应用范围越广, 也越容易被理解。但是, 为了模型拟合好, 我们又期望模型越复杂越好, 解释变量越多, 模型的拟合优度$R^2$
数据介绍
使用上一篇教程用过的数据: icecream.dta, 具体内容可以参看上篇文章: stata教程05-自相关的检验和处理/
1 | use data/icecream.dta, clear |
进行回归分析
1 | reg consumption temp price income |
输出(stream):
Source | SS df MS Number of obs = 30 -------------+---------------------------------- F(3, 26) = 22.17 Model | .090250523 3 .030083508 Prob > F = 0.0000 Residual | .035272835 26 .001356647 R-squared = 0.7190 -------------+---------------------------------- Adj R-squared = 0.6866 Total | .125523358 29 .004328392 Root MSE = .03683 ------------------------------------------------------------------------------ consumption | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- temp | .0034584 .0004455 7.76 0.000 .0025426 .0043743 price | -1.044413 .834357 -1.25 0.222 -2.759458 .6706322 income | .0033078 .0011714 2.82 0.009 .0008999 .0057156 _cons | .1973149 .2702161 0.73 0.472 -.3581223 .752752 ------------------------------------------------------------------------------
Source | SS df MS Number of obs = 30 -------------+---------------------------------- F(3, 26) = 22.17 Model | .090250523 3 .030083508 Prob > F = 0.0000 Residual | .035272835 26 .001356647 R-squared = 0.7190 -------------+---------------------------------- Adj R-squared = 0.6866 Total | .125523358 29 .004328392 Root MSE = .03683 ------------------------------------------------------------------------------ consumption | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- temp | .0034584 .0004455 7.76 0.000 .0025426 .0043743 price | -1.044413 .834357 -1.25 0.222 -2.759458 .6706322 income | .0033078 .0011714 2.82 0.009 .0008999 .0057156 _cons | .1973149 .2702161 0.73 0.472 -.3581223 .752752 ------------------------------------------------------------------------------
计算AIC&BIC
1 | estat ic |
输出(stream):
Akaike's information criterion and Bayesian information criterion ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- . | 30 39.57876 58.61944 4 -109.2389 -103.6341 ----------------------------------------------------------------------------- Note: N=Obs used in calculating BIC; see [R] BIC note.
Akaike's information criterion and Bayesian information criterion ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- . | 30 39.57876 58.61944 4 -109.2389 -103.6341 ----------------------------------------------------------------------------- Note: N=Obs used in calculating BIC; see [R] BIC note.
加入temp的1阶滞后项
1 | reg consumption temp L.temp price income |
输出(stream):
Source | SS df MS Number of obs = 29 -------------+---------------------------------- F(4, 24) = 28.98 Model | .103387183 4 .025846796 Prob > F = 0.0000 Residual | .021406049 24 .000891919 R-squared = 0.8285 -------------+---------------------------------- Adj R-squared = 0.7999 Total | .124793232 28 .004456901 Root MSE = .02987 ------------------------------------------------------------------------------ consumption | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- temp | --. | .0053321 .0006704 7.95 0.000 .0039484 .0067158 L1. | -.0022039 .0007307 -3.02 0.006 -.0037119 -.0006959 | price | -.8383021 .6880205 -1.22 0.235 -2.258307 .5817025 income | .0028673 .0010533 2.72 0.012 .0006934 .0050413 _cons | .1894822 .2323169 0.82 0.423 -.2899963 .6689607 ------------------------------------------------------------------------------
Source | SS df MS Number of obs = 29 -------------+---------------------------------- F(4, 24) = 28.98 Model | .103387183 4 .025846796 Prob > F = 0.0000 Residual | .021406049 24 .000891919 R-squared = 0.8285 -------------+---------------------------------- Adj R-squared = 0.7999 Total | .124793232 28 .004456901 Root MSE = .02987 ------------------------------------------------------------------------------ consumption | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- temp | --. | .0053321 .0006704 7.95 0.000 .0039484 .0067158 L1. | -.0022039 .0007307 -3.02 0.006 -.0037119 -.0006959 | price | -.8383021 .6880205 -1.22 0.235 -2.258307 .5817025 income | .0028673 .0010533 2.72 0.012 .0006934 .0050413 _cons | .1894822 .2323169 0.82 0.423 -.2899963 .6689607 ------------------------------------------------------------------------------
再计算AIC&BIC
1 | estat ic |
输出(stream):
Akaike's information criterion and Bayesian information criterion ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- . | 29 37.85248 63.41576 5 -116.8315 -109.995 ----------------------------------------------------------------------------- Note: N=Obs used in calculating BIC; see [R] BIC note.
Akaike's information criterion and Bayesian information criterion ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- . | 29 37.85248 63.41576 5 -116.8315 -109.995 ----------------------------------------------------------------------------- Note: N=Obs used in calculating BIC; see [R] BIC note.
我们可以看到AIC和BIC都下降了。
再加入temp的2阶滞后项
1 | reg consumption temp L.temp L2.temp price income |
输出(stream):
Source | SS df MS Number of obs = 28 -------------+---------------------------------- F(5, 22) = 21.92 Model | .103722201 5 .02074444 Prob > F = 0.0000 Residual | .020822754 22 .000946489 R-squared = 0.8328 -------------+---------------------------------- Adj R-squared = 0.7948 Total | .124544954 27 .004612776 Root MSE = .03077 ------------------------------------------------------------------------------ consumption | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- temp | --. | .0047858 .0013502 3.54 0.002 .0019856 .007586 L1. | -.0010836 .0022905 -0.47 0.641 -.0058338 .0036666 L2. | -.0008022 .0013414 -0.60 0.556 -.0035841 .0019797 | price | -.7326035 .7214324 -1.02 0.321 -2.228763 .7635558 income | .0026704 .0011308 2.36 0.027 .0003252 .0050156 _cons | .1883478 .23949 0.79 0.440 -.3083241 .6850196 ------------------------------------------------------------------------------
Source | SS df MS Number of obs = 28 -------------+---------------------------------- F(5, 22) = 21.92 Model | .103722201 5 .02074444 Prob > F = 0.0000 Residual | .020822754 22 .000946489 R-squared = 0.8328 -------------+---------------------------------- Adj R-squared = 0.7948 Total | .124544954 27 .004612776 Root MSE = .03077 ------------------------------------------------------------------------------ consumption | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- temp | --. | .0047858 .0013502 3.54 0.002 .0019856 .007586 L1. | -.0010836 .0022905 -0.47 0.641 -.0058338 .0036666 L2. | -.0008022 .0013414 -0.60 0.556 -.0035841 .0019797 | price | -.7326035 .7214324 -1.02 0.321 -2.228763 .7635558 income | .0026704 .0011308 2.36 0.027 .0003252 .0050156 _cons | .1883478 .23949 0.79 0.440 -.3083241 .6850196 ------------------------------------------------------------------------------
再计算AIC&BIC
1 | estat ic |
输出(stream):
Akaike's information criterion and Bayesian information criterion ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- . | 28 36.08382 61.12451 6 -110.249 -102.2558 ----------------------------------------------------------------------------- Note: N=Obs used in calculating BIC; see [R] BIC note.
Akaike's information criterion and Bayesian information criterion ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- . | 28 36.08382 61.12451 6 -110.249 -102.2558 ----------------------------------------------------------------------------- Note: N=Obs used in calculating BIC; see [R] BIC note.
我们可以看到, 在增加了L2项滞后, 我们的AIC和BIC反而上升, 说明增加二阶滞后项导致模型复杂度上升, 但并没有带来模型的拟合优度较大的上升, 也就是得不偿失。
总结
AIC和BIC是两个常用的用户评估模型复杂性的指标, 但是他们略有不同, BIC是一致估计, 而AIC不是, 但现实样本不可能无限大, 而BIC可能导致模型过小, 所以我们通常是综合考虑两个指标。
注意
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