根据 Ender 的 Applied Econometric Time Series ，变量 y 的二阶差分定义为:
Pandas 提供了 diff 函数，它接收“periods”作为参数。尽管如此，df.diff(2) 给出的结果与 df.diff().diff() 不同。
显示上述内容的代码摘录:
In [8]: df
Out[8]:
C.1 C.2 C.3 C.4 C.5 C.6
C.0
1990 16.0 6.0 256.0 216.0 65536 4352
1991 17.0 7.0 289.0 343.0 131072 5202
1992 6.0 -4.0 36.0 -64.0 64 252
1993 7.0 -3.0 49.0 -27.0 128 392
1994 8.0 -2.0 64.0 -8.0 256 576
1995 13.0 3.0 169.0 27.0 8192 2366
1996 10.0 0.5 100.0 0.5 1024 1100
1997 11.0 1.0 121.0 1.0 2048 1452
1998 4.0 -6.0 16.0 -216.0 16 80
1999 5.0 -5.0 25.0 -125.0 32 150
2000 18.0 8.0 324.0 512.0 262144 6156
2001 3.0 -7.0 9.0 -343.0 8 36
2002 0.5 -10.0 0.5 -1000.0 48 20
2003 1.0 -9.0 1.0 -729.0 2 2
2004 14.0 4.0 196.0 64.0 16384 2940
2005 15.0 5.0 225.0 125.0 32768 3600
2006 12.0 2.0 144.0 8.0 4096 1872
2007 9.0 -1.0 81.0 -1.0 512 810
2008 2.0 -8.0 4.0 -512.0 4 12
2009 19.0 9.0 361.0 729.0 524288 7220
In [9]: df.diff(2)
Out[9]:
C.1 C.2 C.3 C.4 C.5 C.6
C.0
1990 NaN NaN NaN NaN NaN NaN
1991 NaN NaN NaN NaN NaN NaN
1992 -10.0 -10.0 -220.0 -280.0 -65472.0 -4100.0
1993 -10.0 -10.0 -240.0 -370.0 -130944.0 -4810.0
1994 2.0 2.0 28.0 56.0 192.0 324.0
1995 6.0 6.0 120.0 54.0 8064.0 1974.0
1996 2.0 2.5 36.0 8.5 768.0 524.0
1997 -2.0 -2.0 -48.0 -26.0 -6144.0 -914.0
1998 -6.0 -6.5 -84.0 -216.5 -1008.0 -1020.0
1999 -6.0 -6.0 -96.0 -126.0 -2016.0 -1302.0
2000 14.0 14.0 308.0 728.0 262128.0 6076.0
2001 -2.0 -2.0 -16.0 -218.0 -24.0 -114.0
2002 -17.5 -18.0 -323.5 -1512.0 -262096.0 -6136.0
2003 -2.0 -2.0 -8.0 -386.0 -6.0 -34.0
2004 13.5 14.0 195.5 1064.0 16336.0 2920.0
2005 14.0 14.0 224.0 854.0 32766.0 3598.0
2006 -2.0 -2.0 -52.0 -56.0 -12288.0 -1068.0
2007 -6.0 -6.0 -144.0 -126.0 -32256.0 -2790.0
2008 -10.0 -10.0 -140.0 -520.0 -4092.0 -1860.0
2009 10.0 10.0 280.0 730.0 523776.0 6410.0
In [10]: df.diff().diff()
Out[10]:
C.1 C.2 C.3 C.4 C.5 C.6
C.0
1990 NaN NaN NaN NaN NaN NaN
1991 NaN NaN NaN NaN NaN NaN
1992 -12.0 -12.0 -286.0 -534.0 -196544.0 -5800.0
1993 12.0 12.0 266.0 444.0 131072.0 5090.0
1994 0.0 0.0 2.0 -18.0 64.0 44.0
1995 4.0 4.0 90.0 16.0 7808.0 1606.0
1996 -8.0 -7.5 -174.0 -61.5 -15104.0 -3056.0
1997 4.0 3.0 90.0 27.0 8192.0 1618.0
1998 -8.0 -7.5 -126.0 -217.5 -3056.0 -1724.0
1999 8.0 8.0 114.0 308.0 2048.0 1442.0
2000 12.0 12.0 290.0 546.0 262096.0 5936.0
2001 -28.0 -28.0 -614.0 -1492.0 -524248.0 -12126.0
2002 12.5 12.0 306.5 198.0 262176.0 6104.0
2003 3.0 4.0 9.0 928.0 -86.0 -2.0
2004 12.5 12.0 194.5 522.0 16428.0 2956.0
2005 -12.0 -12.0 -166.0 -732.0 2.0 -2278.0
2006 -4.0 -4.0 -110.0 -178.0 -45056.0 -2388.0
2007 0.0 0.0 18.0 108.0 25088.0 666.0
2008 -4.0 -4.0 -14.0 -502.0 3076.0 264.0
2009 24.0 24.0 434.0 1752.0 524792.0 8006.0
In [11]: df.diff(2) - df.diff().diff()
Out[11]:
C.1 C.2 C.3 C.4 C.5 C.6
C.0
1990 NaN NaN NaN NaN NaN NaN
1991 NaN NaN NaN NaN NaN NaN
1992 2.0 2.0 66.0 254.0 131072.0 1700.0
1993 -22.0 -22.0 -506.0 -814.0 -262016.0 -9900.0
1994 2.0 2.0 26.0 74.0 128.0 280.0
1995 2.0 2.0 30.0 38.0 256.0 368.0
1996 10.0 10.0 210.0 70.0 15872.0 3580.0
1997 -6.0 -5.0 -138.0 -53.0 -14336.0 -2532.0
1998 2.0 1.0 42.0 1.0 2048.0 704.0
1999 -14.0 -14.0 -210.0 -434.0 -4064.0 -2744.0
2000 2.0 2.0 18.0 182.0 32.0 140.0
2001 26.0 26.0 598.0 1274.0 524224.0 12012.0
2002 -30.0 -30.0 -630.0 -1710.0 -524272.0 -12240.0
2003 -5.0 -6.0 -17.0 -1314.0 80.0 -32.0
2004 1.0 2.0 1.0 542.0 -92.0 -36.0
2005 26.0 26.0 390.0 1586.0 32764.0 5876.0
2006 2.0 2.0 58.0 122.0 32768.0 1320.0
2007 -6.0 -6.0 -162.0 -234.0 -57344.0 -3456.0
2008 -6.0 -6.0 -126.0 -18.0 -7168.0 -2124.0
2009 -14.0 -14.0 -154.0 -1022.0 -1016.0 -1596.0
为什么不同？哪一个对应于安德书中定义的那个？
请您参考如下方法:
正是因为
Δ2 yt = yt - 2 yt - 1 + y< sub>t - 2 ≠ yt - yt - 2。
左侧是 df.diff().diff()，而右侧是 df.diff(2)。对于差异中的差异，您想要左侧。
