Confidence interval for weighted average

[Wen-Hung Hsu]

Dear all,

I would love seek for help here. What I am doing is that, I want to get the confidence interval for a weighted average.
This is how the weighting method ( variance-weighted). First, get the proportion of the variance for each group, and sum the product of the share part and the weights.

egen total_var = total(var_RET)
gen var_weight = var_RET/ total_var
gen mktinfo_share_weight = mktinfo_share * var_weight
egen mktinfo_share_weighted = sum(mktinfo_share_weight)

For common command -ci-, since I only have one final weighted average across the rows, if I type
[CODE]
ci means mktinfo_share_weighted , level(99)
[CODE]
it cannot gives me the correct estimation for the interval.
Therefore, I would love to ask if there is another way to show the confident interval for such weighted average.
The idea may derive from the original definition of CI:
mean +- t-value * standard deviation/ root of observation
As long as I can figure out the part after +-, I can re-create the confident interval.

* Example generated by -dataex-. For more info, type help dataex
clear
input float(id var_RET mktinfo)
  1 .00025453698  .00009306284
  2  .0010932052 1.6395407e-12
  3 .00013873918 .000015883144
  4 .00016952788  .00005152942
  5 .00015983787 .000065545595
  6 .00024326568  .00002890711
  7 .00016386366 .000017440707
  8 .00028597284  .00011248127
  9 .00023394095  .00008871407
 10 .00034012095  3.477177e-06
 11  .0007160784  .00013231797
 12  .0004039086  .00002951982
 13   .000220585 .000011787646
 14 .00010360737 .000011535892
 15 .00012653777  .00002533065
 16  .0003179252  .00002351869
 17 .00017056165 .000016126445
 18  .0005434161  .00006128963
 19 .00010800704  1.460614e-06
 20 .00020800895  .00007542817
 21 .00023145023  .00003084261
 22   .000695263  .00008889187
 23 .00019090103 .000018818724
 24 .00022233406 .000034568246
 25 .00009257844 .000011349546
 26  .0004938965  2.323115e-06
 27 .00050838693  .00001416657
 28  .0003364799  .00011926446
 29 .00027392787 3.8549214e-07
 30  .0003095213 .000072238654
 31  .0002522189  .00005137531
 32 .00015967217 .000017764502
 33   .000359111  .00003512886
 34  .0004625673  .00002947951
 35  .0001509002  6.536637e-06
 36  .0002188882 .000035863814
 37 .00023711416 .000018214803
 38 .00015291903  7.146552e-06
 39 .00015346557  .00002819447
 40 .00053468865  .00005954452
 41  .0003625301 .000026964013
 42 .00023529546  .00013698229
 43 .00023128066 .000012145727
 44  .0002992186  .00003733501
 45 .00045417435  .00002429819
 46 .00023781646   .0000285013
 47 .00016155763  .00003025555
 48  .0009921843 1.6805366e-06
 49  .0001302785 .000011270977
 50 .00004920922  8.644625e-06
 51  .0005262289  .00002889953
 52  .0003162579  .00005325446
 53  .0004438201  .00005493474
 54  .0006693757  .00004698702
 56 .00011338643   4.96667e-06
 57 .00015950597  .00008380983
 58  .0011968252  .00013898936
 59 .00040132646  .00004648703
 60  .0005500457   .0001222533
 61 .00035834225 .000013325222
 62  .0006459293  .00005520241
 63  .0004001613  .00008322814
 64  .0004163074   .0000336237
 65  .0011161747  .00004507854
 66  .0004083046   .0000481796
 67 .00011901466  .00005155581
 68  .0004205853  .00010002564
 69  .0006257504 .000027802735
 70  .0003540163 .000031119293
 71 .00028855423 .000062650455
 72    .00020876  .00001857438
 73 .00011886151 .000012743667
 74   .001184069   .0000846461
 75 .00008185631   .0000181655
 76 .00017324876  .00005542144
 77 .00010932072  3.303772e-06
 78  .0004687891  .00003822233
 79 .00026965962 .000020909243
 80 .00005452983  3.757928e-06
 81  .0002793335  .00006898495
 82 .00017664494  .00005756211
 83  .0001835153  .00008233645
 84  .0003393123 .000027126114
 85 .00022325464 .000019327326
 86   .000078872 .000017647546
 87  .0004393946  .00001253071
 88  .0005859645  .00006841045
 89  .0004759088  .00007075154
 90 .00031620075  .00005308945
 91 .00025006154  .00004220495
 92  .0007614839   .0001366144
 93 .00005937216  .00001791885
 94 .00044717905   9.66545e-07
 95 .00009891028   .0000674664
 96  .0004589452  .00013130865
 97 .00019592948   .0001016715
 98 .00027104112  .00008730995
 99   .000129755  7.287784e-06
100  .0003375435   .0000227227
101  .0010927363  .00020377597
end

Forget to mention, the id is the group of stock-year observation. The variance is firstly calculated within each id, and then the id is duplicated drop to leave only 1 observation for each id.

[Wen-Hung Hsu]

There is second question I would love to post:
Also for the weighted average, how can I perform t-test of difference in means between different period within the same variable?
For example,

gen sub = 1 
replace sub = 2 if year >1996
bysort sub: egen total_var_B = total(var_RET)
gen var_weight_B = var_RET/ total_var_B
gen mktinfo_share_weight_B = mktinfo_share_w * var_weight_B
bysort sub: egen mktinfo_share_B = sum(mktinfo_share_weight_B)

Since now the weighted average is still, only one value within sub-period. How can I perform the t-test for the difference in means between two sub-period?

[Clyde Schechter]

Re #1:

mean mktinfo [aweight = var_RET]

Re #2:

regress mktinfo i.sub [aweight =var_RET]

and the difference and its associated standard error, CIs and test statistics are shown in the 2.sub row in the output table.
Note: The code shown for #2 is untested because the example data does not include a year variable.

Added: General principle: in Stata, calculations involving weighted means usually are done with the unweighted data and a reference in the command to the desired weighting. While it is fine to calculated weighted means as new variables, they usually don't play a role in weighted analyses of the data.