Hello!
I am using Stata 19, and I am a having a tough time interpreting my correlated random effects output, especially in light of what is indicated by my descriptive statistics, and my RE estimates (xtreg, re), which the Mundlak test indicates to be efficient.
My primary independent variable of interest is TS_ce, a categorical indicator that distinguishes between six different excise tax structure groupings. I am interested in understanding how different tax structures influence price dispersion, which is the share of cheapest brand price in premium brand price (the higher the % the smaller the gap). I have an unbalanced panel of country-level data: 910 observations from 180 countries. Data are available every two years over the period 2014-2024 (i.e., T=6).
My data are xtset, where id= country, as follows:
This is what my data look like:
In principle, my research question can be answered by looking at variation in the tax structure within countries over time; or by allowing differences within and between countries to factor in. My literature search indicated the need for some time invariant predictors, so xtreg, cre seemed a sensible starting point. Below is the output. From the Mundlak test, I fail to reject the null that random effects are efficient. So this provides some support for going with xtreg, re. However, I’ve observed that most of the very experienced people on this forum don’t believe that hypothesis testing should be the sole basis for selecting one estimator over another. So, I thought it would sensible to report the results of the Mundlak test, and then report results from both the RE and CRE models.
However, on my understanding that the xt_means shown in the CRE output show the “between effects”; I am not sure how what to make of these results for two reasons. First, at a descriptive level, variation between countries drives variation in tax structure (TS_ce) to a much greater extent than variation within countries over time (please see xtsum output below); so I am surprised to see that most of the xt_means for TS_ce are not statistically different from zero.
Second, I am finding it very hard to reconcile these CRE results with my RE estimates (shown below). While I can fully appreciate that “xtreg, re” is a different estimation procedure; I am left stuck on how to interpret my CRE model output in light of the RE estimates. The RE estimates are telling me a story that makes sense. All tax structures other than a uniform specific tax (the base of TS_ce) decrease my measure of price dispersion (and a reduction in this % indicates a larger gap between the most expensive and cheapest brand). My understanding of what the CRE model is telling me, by contrast, is that when I control for unobserved time-invariant country-level factors; the between effect (xtmeans) is only statistically different from zero for countries that adopt a mixed tiered structure. Is this simply evidence that my RE results aren't robust to a CRE estimation? Or is the estimation of the CRE model simply not a smart way to try and model the relationship.
Thank you for taking the time to read this!
Sam
I am using Stata 19, and I am a having a tough time interpreting my correlated random effects output, especially in light of what is indicated by my descriptive statistics, and my RE estimates (xtreg, re), which the Mundlak test indicates to be efficient.
My primary independent variable of interest is TS_ce, a categorical indicator that distinguishes between six different excise tax structure groupings. I am interested in understanding how different tax structures influence price dispersion, which is the share of cheapest brand price in premium brand price (the higher the % the smaller the gap). I have an unbalanced panel of country-level data: 910 observations from 180 countries. Data are available every two years over the period 2014-2024 (i.e., T=6).
My data are xtset, where id= country, as follows:
Code:
xtset id year, delta(2)
Code:
* Example generated by -dataex-. For more info, type help dataex clear input float id double price_dispersion_use float TS_ce byte POWE double unem float(income region_id) int year 1 44.44444444444444 1 18 18.055 3 4 2014 1 56.666666666666664 1 18 15.418 3 4 2016 1 62.5 1 18 12.304 3 4 2018 1 60.60606060606061 1 19 11.69 3 4 2020 1 60 1 19 10.137 3 4 2022 1 50 1 19 10.25 3 4 2024 2 33.33333333333333 4 12 10.207 3 1 2014 2 35.714285714285715 6 12 10.202 3 1 2016 2 15 6 13 12.137 3 1 2018 2 50 6 13 14.057 3 1 2020 2 48.57142857142857 3 13 12.346 3 1 2022 2 42.857142857142854 6 13 11.427 3 1 2024 3 72.85714285714285 4 11 5.3 1 4 2014 3 72.85714285714285 4 11 3.3 1 4 2016 3 77.77777777777777 1 11 1.8 1 4 2018 3 68.44993141289439 1 11 2.9 1 4 2020 3 69.86301369863014 1 11 2.1 1 4 2022 3 59.09090909090908 1 11 1.4 1 4 2024 4 25 2 13 16.69 3 1 2020 4 25 2 13 14.602 3 1 2022 4 28.57142857142857 2 13 14.464 3 1 2024 5 40 2 16 7.423938916311391 1 2 2024 6 41.935483870967744 2 18 7.268 3 2 2014 6 37.75 2 18 8.085 3 2 2016 6 45.34920634920635 2 18 9.22 3 2 2018 6 18.726114649681527 2 18 11.461 3 2 2020 6 13.384615384615383 2 18 6.805 3 2 2022 6 40.055248618784525 2 18 7.876 3 2 2024 7 30 4 13 17.498 3 4 2014 7 26.666666666666668 4 14 17.617 3 4 2016 7 42.857142857142854 2 14 18.966 3 4 2018 7 47.5 1 18 18.175 3 4 2020 7 48.23529411764706 1 19 13.379 3 4 2022 7 35 1 19 13.329 3 4 2024 8 78.93318965517241 1 19 6.078 1 6 2014 8 73.84341637010677 1 19 5.711 1 6 2016 8 82.34126984126985 1 19 5.3 1 6 2018 8 71.02189781021899 4 18 6.456 1 6 2020 8 68.45524542829644 4 18 3.728 1 6 2022 8 60.58098915241773 4 18 4.072 1 6 2024 9 80.61224489795919 3 13 5.674 1 4 2014 9 80 3 15 6.064 1 4 2016 9 80 3 15 4.933 1 4 2018 9 82.45614035087719 3 17 5.201 1 4 2020 9 68.35820895522387 3 16 4.992 1 4 2022 9 83.07692307692308 3 16 5.439 1 4 2024 10 24 4 12 4.91 3 4 2014 10 56.25 1 13 5 3 4 2016 10 23.076923076923077 4 14 4.9 3 4 2018 10 47.05882352941177 4 14 7.24 3 4 2020 10 55.55555555555556 4 14 5.65 3 4 2022 10 48.88888888888889 1 14 5.594 3 4 2024 11 48.658536585365916 1 10 13.8 1 2 2014 11 40.22346368715088 1 10 12.7 1 2 2016 11 61.08949416342412 1 10 12.027 1 2 2020 11 81.76100628930817 1 10 8.463 1 2 2024 12 40 2 15 1.2 1 3 2018 12 34.78260869565218 2 15 1.781 1 3 2020 12 58.333333333333336 2 15 1.326 1 3 2022 12 58.333333333333336 2 15 1.102 1 3 2024 13 15.789473684210526 5 13 4.416 3 5 2014 13 22.727272727272727 5 15 4.35 3 5 2016 13 33.33333333333333 5 15 4.407 3 5 2018 13 32.142857142857146 5 15 5.436 3 5 2020 13 25 5 15 4.593 3 5 2022 13 30.864197530864196 5 15 4.68 3 5 2024 14 79.32850559578671 1 13 12.17 1 2 2014 14 81.25 1 13 8.247 1 2 2016 14 45.23433385992628 1 16 8.322 1 2 2018 14 78.84615384615384 1 16 8.365 1 2 2022 14 79.98999499749875 1 15 7.529 1 2 2024 15 35.625 4 14 5.902 3 4 2014 15 31.914893617021278 4 15 5.844 3 4 2016 15 30.645161290322577 4 15 4.763 3 4 2018 15 25.71428571428572 4 15 4.049 3 4 2020 15 23.958333333333332 4 15 3.574 3 4 2022 15 47.26027397260275 4 15 3.361 3 4 2024 16 80.82901554404145 3 14 8.523 1 4 2014 16 81.64556962025317 3 15 7.83 1 4 2016 16 83.33333333333334 3 15 5.941 1 4 2018 16 85.29411764705883 3 15 5.545 1 4 2020 16 72.5 3 15 5.57 1 4 2022 16 65.21739130434783 3 15 5.488 1 4 2024 17 41.66666666666667 1 10 7 3 2 2016 17 40 1 10 7.896 3 2 2018 17 50 1 10 10.784 3 2 2020 17 50.391644908616186 1 10 8.763 3 2 2022 17 55.55555555555556 1 10 7 3 2 2024 18 20 2 15 1.784 3 1 2016 18 22.5 2 15 1.41 3 1 2018 18 47.368421052631575 2 15 1.502 3 1 2020 18 50 2 17 1.722 3 1 2024 19 35.714285714285715 2 16 2.021 3 2 2014 19 24 5 16 3.498 3 2 2016 19 40 4 14 3.519 3 2 2018 19 50 4 18 3.552 3 2 2022 19 36.40776699029126 4 16 3.091 3 2 2024 20 67.44186046511628 3 10 27.517 3 4 2014 20 60.416666666666664 3 10 25.408 3 4 2016 20 68.96551724137932 3 10 18.4 3 4 2018 end label values TS_ce TS_ce_l label def TS_ce_l 1 "specific uniform", modify label def TS_ce_l 2 "advalorem uniform", modify label def TS_ce_l 3 "mixed uniform", modify label def TS_ce_l 4 "specific_tiered", modify label def TS_ce_l 5 "advalorem tiered", modify label def TS_ce_l 6 "mixed tiered", modify label values region_id region_id_l label def region_id_l 1 "AFR", modify label def region_id_l 2 "AMR", modify label def region_id_l 3 "EMR", modify label def region_id_l 4 "EUR", modify label def region_id_l 5 "SEA", modify label def region_id_l 6 "WPR", modify
Code:
. xtreg price_dispersion_use i.TS_ce POWE unem i.income i. region_id i.year, cre vce(cluster id)
note: 2.income omitted from xt_means because of collinearity.
note: 3.income omitted from xt_means because of collinearity.
note: 2.region_id omitted from xt_means because of collinearity.
note: 3.region_id omitted from xt_means because of collinearity.
note: 4.region_id omitted from xt_means because of collinearity.
note: 5.region_id omitted from xt_means because of collinearity.
note: 6.region_id omitted from xt_means because of collinearity.
note: 2014b.year omitted from xt_means because of collinearity.
Correlated random-effects regression Number of obs = 910
Group variable: id Number of groups = 180
R-squared: Obs per group:
Within = 0.0846 min = 1
Between = 0.6160 avg = 5.1
Overall = 0.4941 max = 6
Wald chi2(19) = 174.94
corr(xit_vars*b, xt_means*γ) = -0.0234 Prob > chi2 = 0.0000
(Std. err. adjusted for 180 clusters in id)
------------------------------------------------------------------------------------
| Robust
price_dispersion~e | Coefficient std. err. z P>|z| [95% conf. interval]
-------------------+----------------------------------------------------------------
xit_vars |
TS_ce |
advalorem uniform | -7.00536 5.691736 -1.23 0.218 -18.16096 4.150238
mixed uniform | 6.233715 5.841014 1.07 0.286 -5.214461 17.68189
specific_tiered | -9.882044 4.106249 -2.41 0.016 -17.93015 -1.833944
advalorem tiered | -9.838445 6.437943 -1.53 0.126 -22.45658 2.779692
mixed tiered | -4.556287 6.71061 -0.68 0.497 -17.70884 8.596268
|
POWE | .3949454 .6024939 0.66 0.512 -.7859209 1.575812
unem | -.5789714 .3027195 -1.91 0.056 -1.172291 .014348
|
income |
2 | -21.24649 4.182089 -5.08 0.000 -29.44324 -13.04975
3 | -14.13742 2.842535 -4.97 0.000 -19.70869 -8.566158
|
region_id |
AMR | 5.557842 4.096317 1.36 0.175 -2.470792 13.58648
EMR | -1.54277 4.222431 -0.37 0.715 -9.818583 6.733043
EUR | 10.3946 4.720081 2.20 0.028 1.143407 19.64579
SEA | -5.420182 6.495625 -0.83 0.404 -18.15137 7.31101
WPR | 5.069165 5.071582 1.00 0.318 -4.870953 15.00928
|
year |
2016 | 1.026464 1.19546 0.86 0.391 -1.316595 3.369524
2018 | 1.466876 1.592039 0.92 0.357 -1.653463 4.587215
2020 | 2.612579 1.628073 1.60 0.109 -.5783851 5.803542
2022 | 2.105276 1.735296 1.21 0.225 -1.295843 5.506394
2024 | 3.114156 1.929465 1.61 0.107 -.6675257 6.895838
|
_cons | 61.47376 13.28127 4.63 0.000 35.44295 87.50458
-------------------+----------------------------------------------------------------
xt_means |
TS_ce |
advalorem uniform | -8.170804 6.891239 -1.19 0.236 -21.67738 5.335775
mixed uniform | -9.508535 6.53317 -1.46 0.146 -22.31331 3.296242
specific_tiered | -2.998356 6.78168 -0.44 0.658 -16.2902 10.29349
advalorem tiered | -7.560736 10.26701 -0.74 0.461 -27.6837 12.56222
mixed tiered | -26.60391 10.31528 -2.58 0.010 -46.82148 -6.386337
|
POWE | .0419637 .7152601 0.06 0.953 -1.35992 1.443848
unem | .5155043 .3608912 1.43 0.153 -.1918295 1.222838
|
income |
2 | 0 (omitted)
3 | 0 (omitted)
|
region_id |
AMR | 0 (omitted)
EMR | 0 (omitted)
EUR | 0 (omitted)
SEA | 0 (omitted)
WPR | 0 (omitted)
|
year |
2016 | 3.785794 15.98809 0.24 0.813 -27.55028 35.12187
2018 | 12.67214 17.29928 0.73 0.464 -21.23382 46.57811
2020 | -14.10317 15.83415 -0.89 0.373 -45.13754 16.9312
2022 | -3.180658 13.25776 -0.24 0.810 -29.16539 22.80407
2024 | -5.024841 13.70644 -0.37 0.714 -31.88897 21.83929
-------------------+----------------------------------------------------------------
sigma_u | 12.287152
sigma_e | 11.679675
rho | .52533028 (fraction of variance due to u_i)
------------------------------------------------------------------------------------
Mundlak test (xt_means = 0): chi2(12) = 13.1329 Prob > chi2 = 0.3595
. estat mundlak
Mundlak specification test
H0: Covariates are uncorrelated with unobserved panel-level effects
chi2(12) = 13.13
Prob > chi2 = 0.3595
Notes: Fixed effects and correlated random effects are
consistent under H0 and Ha.
Random effects are efficient under H0.
However, on my understanding that the xt_means shown in the CRE output show the “between effects”; I am not sure how what to make of these results for two reasons. First, at a descriptive level, variation between countries drives variation in tax structure (TS_ce) to a much greater extent than variation within countries over time (please see xtsum output below); so I am surprised to see that most of the xt_means for TS_ce are not statistically different from zero.
Code:
. xtsum TS_ce
Variable | Mean Std. dev. Min Max | Observations
-----------------+--------------------------------------------+----------------
TS_ce overall | 2.584615 1.316304 1 6 | N = 910
between | 1.134847 1 6 | n = 180
within | .6594633 -.4153846 6.251282 | T-bar = 5.05556
Second, I am finding it very hard to reconcile these CRE results with my RE estimates (shown below). While I can fully appreciate that “xtreg, re” is a different estimation procedure; I am left stuck on how to interpret my CRE model output in light of the RE estimates. The RE estimates are telling me a story that makes sense. All tax structures other than a uniform specific tax (the base of TS_ce) decrease my measure of price dispersion (and a reduction in this % indicates a larger gap between the most expensive and cheapest brand). My understanding of what the CRE model is telling me, by contrast, is that when I control for unobserved time-invariant country-level factors; the between effect (xtmeans) is only statistically different from zero for countries that adopt a mixed tiered structure. Is this simply evidence that my RE results aren't robust to a CRE estimation? Or is the estimation of the CRE model simply not a smart way to try and model the relationship.
Code:
. xtreg price_dispersion_use i.TS_ce POWE unem i.income i.region_id i.year, re vce(cluster id)
Random-effects GLS regression Number of obs = 910
Group variable: id Number of groups = 180
R-squared: Obs per group:
Within = 0.0765 min = 1
Between = 0.5851 avg = 5.1
Overall = 0.4752 max = 6
Wald chi2(19) = 474.68
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. err. adjusted for 180 clusters in id)
------------------------------------------------------------------------------------
| Robust
price_dispersion~e | Coefficient std. err. z P>|z| [95% conf. interval]
-------------------+----------------------------------------------------------------
TS_ce |
advalorem uniform | -12.68268 3.205268 -3.96 0.000 -18.96489 -6.400475
mixed uniform | -.1250515 3.034091 -0.04 0.967 -6.07176 5.821657
specific_tiered | -12.0051 3.283801 -3.66 0.000 -18.44123 -5.568966
advalorem tiered | -14.78059 4.070479 -3.63 0.000 -22.75859 -6.8026
mixed tiered | -12.34087 4.762831 -2.59 0.010 -21.67585 -3.005894
|
POWE | .3580088 .346207 1.03 0.301 -.3205445 1.036562
unem | -.2119297 .1624692 -1.30 0.192 -.5303634 .106504
|
income |
2 | -21.74245 3.636952 -5.98 0.000 -28.87075 -14.61416
3 | -15.02823 2.638187 -5.70 0.000 -20.19898 -9.857476
|
region_id |
AMR | 7.313431 3.643355 2.01 0.045 .1725851 14.45428
EMR | -3.950831 4.557214 -0.87 0.386 -12.88281 4.981144
EUR | 9.989994 4.205076 2.38 0.018 1.748197 18.23179
SEA | -6.8663 5.576326 -1.23 0.218 -17.7957 4.063098
WPR | 6.262471 5.043423 1.24 0.214 -3.622456 16.1474
|
year |
2016 | 1.125533 1.138146 0.99 0.323 -1.105191 3.356258
2018 | 1.80515 1.446099 1.25 0.212 -1.029151 4.639451
2020 | 2.504381 1.458252 1.72 0.086 -.3537408 5.362502
2022 | 2.474173 1.494605 1.66 0.098 -.4551981 5.403544
2024 | 3.501714 1.726538 2.03 0.043 .1177607 6.885667
|
_cons | 60.35417 6.010714 10.04 0.000 48.57339 72.13495
-------------------+----------------------------------------------------------------
sigma_u | 12.287152
sigma_e | 11.679675
rho | .52533028 (fraction of variance due to u_i)
------------------------------------------------------------------------------------
Thank you for taking the time to read this!
Sam
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