In [ ]:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import statsmodels.formula.api as smf
In [ ]:
np.random.seed(123)
numproducts = 1000
# products have a normal distribution of quality
quality = np.random.normal(loc=100, scale=25,size=numproducts)
# price is a positive function of quality
price = np.sqrt(quality) + np.random.normal(loc=10, scale=2, size=numproducts)
# quantity sold and price often have a log-log relationship so that a percentage increase in price
# drives a percentage increase in quantity sold
log_price = np.log(price)
log_quality = np.log(quality)
# with this formula, we know the exact relationship between price, quality, and sales
log_qty_sold = np.maximum(0, (2*log_quality -0.5*log_price))
# typically, your data set will only include price and quantity, and you'll have to create the log transformations
qty_sold = np.exp(log_qty_sold).round()
df = pd.DataFrame({'quality':quality, 'price':price, 'qty_sold':qty_sold, 'log_quality':log_quality,
'log_price':log_price, 'log_qty_sold':log_qty_sold})
df.describe()
Out[Â ]:
| quality | price | qty_sold | log_quality | log_price | log_qty_sold | |
|---|---|---|---|---|---|---|
| count | 1000.000000 | 1000.000000 | 1000.000000 | 1000.000000 | 1000.000000 | 1000.000000 |
| mean | 99.010897 | 19.881620 | 2316.588000 | 4.559104 | 2.982951 | 7.626732 |
| std | 25.032208 | 2.285401 | 1061.261054 | 0.281329 | 0.118414 | 0.532804 |
| min | 19.223625 | 10.682313 | 89.000000 | 2.956140 | 2.368589 | 4.493443 |
| 25% | 82.886085 | 18.340245 | 1558.500000 | 4.417467 | 2.909098 | 7.351315 |
| 50% | 98.970361 | 19.944195 | 2179.000000 | 4.594820 | 2.992938 | 7.686764 |
| 75% | 116.721644 | 21.421391 | 2977.500000 | 4.759792 | 3.064390 | 7.998851 |
| max | 189.289480 | 26.772193 | 7921.000000 | 5.243277 | 3.287364 | 8.977303 |
In [ ]:
aggvars = list(['price','quality','qty_sold'])
df['treated'] = np.random.choice([0,1],size=len(df))
df.groupby('treated')[aggvars].agg(['mean','std'])
Out[Â ]:
| price | quality | qty_sold | ||||
|---|---|---|---|---|---|---|
| mean | std | mean | std | mean | std | |
| treated | ||||||
| 0 | 19.793992 | 2.369162 | 97.311904 | 24.740904 | 2243.255020 | 1033.106130 |
| 1 | 19.968550 | 2.198072 | 100.696352 | 25.229541 | 2389.336653 | 1084.606188 |
In [ ]:
# apply the treatment -- example in percentage: control for quality confounder
discount = 0.10
df['price_experiment'] = np.where(df['treated']==1, df['price']*(1-discount), df['price'])
df['log_price_experiment'] = np.log(df['price_experiment'])
df['log_qty_sold_experiment'] = np.maximum(0, (2*df['log_quality'] -0.5*df['log_price_experiment']))
formula = 'log_qty_sold_experiment ~ log_quality + treated '
model = smf.ols(formula=formula, data=df).fit()
print(model.summary())
OLS Regression Results
===================================================================================
Dep. Variable: log_qty_sold_experiment R-squared: 0.991
Model: OLS Adj. R-squared: 0.991
Method: Least Squares F-statistic: 5.729e+04
Date: Sun, 01 Dec 2024 Prob (F-statistic): 0.00
Time: 18:56:24 Log-Likelihood: 1583.4
No. Observations: 1000 AIC: -3161.
Df Residuals: 997 BIC: -3146.
Df Model: 2
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
Intercept -0.9700 0.026 -37.959 0.000 -1.020 -0.920
log_quality 1.8857 0.006 336.408 0.000 1.875 1.897
treated 0.0516 0.003 16.369 0.000 0.045 0.058
==============================================================================
Omnibus: 74.583 Durbin-Watson: 2.059
Prob(Omnibus): 0.000 Jarque-Bera (JB): 110.589
Skew: 0.580 Prob(JB): 9.68e-25
Kurtosis: 4.144 Cond. No. 78.2
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [ ]:
# apply the treatment -- example in percentage: missing quality confounder
discount = 0.10
df['price_experiment'] = np.where(df['treated']==1, df['price']*(1-discount), df['price'])
df['log_price_experiment'] = np.log(df['price_experiment'])
df['log_qty_sold_experiment'] = np.maximum(0, (2*df['log_quality'] -0.5*df['log_price_experiment']))
formula = 'log_qty_sold_experiment ~ treated '
model = smf.ols(formula=formula, data=df).fit()
print(model.summary())
OLS Regression Results
===================================================================================
Dep. Variable: log_qty_sold_experiment R-squared: 0.012
Model: OLS Adj. R-squared: 0.011
Method: Least Squares F-statistic: 12.28
Date: Sun, 01 Dec 2024 Prob (F-statistic): 0.000478
Time: 18:56:24 Log-Likelihood: -786.96
No. Observations: 1000 AIC: 1578.
Df Residuals: 998 BIC: 1588.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 7.5940 0.024 318.504 0.000 7.547 7.641
treated 0.1179 0.034 3.504 0.000 0.052 0.184
==============================================================================
Omnibus: 209.350 Durbin-Watson: 1.943
Prob(Omnibus): 0.000 Jarque-Bera (JB): 529.483
Skew: -1.097 Prob(JB): 1.06e-115
Kurtosis: 5.810 Cond. No. 2.62
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [ ]:
# add pre-period to estimate diff-in-diff
scrap = df.reset_index()
scrap.rename(columns={'index':'product_id'},inplace=True)
post=scrap.copy()
post['post'] = 1
pre = scrap.copy()
pre['post'] = 0
pre['price_experiment'] = pre['price']
pre['log_price_experiment'] = np.log(pre['price_experiment'])
pre['log_qty_sold_experiment'] = np.maximum(0, (2*pre['log_quality'] -0.5*pre['log_price_experiment']))
combined = pd.concat([pre, post], axis=0)
combined['treated_post'] = combined['treated']*combined['post']
formula = 'log_qty_sold_experiment ~ treated + post + treated_post'
model = smf.ols(formula, data=combined).fit()
print(model.summary())
OLS Regression Results
===================================================================================
Dep. Variable: log_qty_sold_experiment R-squared: 0.009
Model: OLS Adj. R-squared: 0.007
Method: Least Squares F-statistic: 5.758
Date: Sun, 01 Dec 2024 Prob (F-statistic): 0.000641
Time: 18:56:24 Log-Likelihood: -1573.9
No. Observations: 2000 AIC: 3156.
Df Residuals: 1996 BIC: 3178.
Df Model: 3
Covariance Type: nonrobust
================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------
Intercept 7.5940 0.024 318.504 0.000 7.547 7.641
treated 0.0652 0.034 1.939 0.053 -0.001 0.131
post 3.935e-16 0.034 1.17e-14 1.000 -0.066 0.066
treated_post 0.0527 0.048 1.107 0.268 -0.041 0.146
==============================================================================
Omnibus: 410.959 Durbin-Watson: 1.943
Prob(Omnibus): 0.000 Jarque-Bera (JB): 1058.967
Skew: -1.097 Prob(JB): 1.12e-230
Kurtosis: 5.810 Cond. No. 6.87
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [ ]:
# add fixed effects and cluster by product ID
formula = 'log_qty_sold_experiment ~ C(product_id) + post + treated_post - 1'
model = smf.ols(formula, data=combined).fit(covtype='cluster', cov_kwds={'groups':combined['product_id']})
print(model.summary())
print("Treated Post Coefficient: {:.4f}".format(model.params['treated_post']))
print("Treated Post P-value: {:.4f}" .format(model.pvalues['treated_post']))
OLS Regression Results
===================================================================================
Dep. Variable: log_qty_sold_experiment R-squared: 1.000
Model: OLS Adj. R-squared: 1.000
Method: Least Squares F-statistic: 4.818e+24
Date: Mon, 02 Dec 2024 Prob (F-statistic): 0.00
Time: 18:26:44 Log-Likelihood: 55255.
No. Observations: 2000 AIC: -1.085e+05
Df Residuals: 998 BIC: -1.029e+05
Df Model: 1001
Covariance Type: nonrobust
======================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------
C(product_id)[0] 7.1593 2.43e-13 2.94e+13 0.000 7.159 7.159
C(product_id)[1] 8.1030 2.43e-13 3.33e+13 0.000 8.103 8.103
C(product_id)[2] 7.8065 2.43e-13 3.21e+13 0.000 7.806 7.806
C(product_id)[3] 6.8822 2.43e-13 2.83e+13 0.000 6.882 6.882
C(product_id)[4] 7.3951 2.43e-13 3.04e+13 0.000 7.395 7.395
C(product_id)[5] 8.4523 2.43e-13 3.47e+13 0.000 8.452 8.452
C(product_id)[6] 5.9620 2.43e-13 2.45e+13 0.000 5.962 5.962
C(product_id)[7] 7.5616 2.43e-13 3.11e+13 0.000 7.562 7.562
C(product_id)[8] 8.2660 2.43e-13 3.4e+13 0.000 8.266 8.266
C(product_id)[9] 7.3264 2.43e-13 3.01e+13 0.000 7.326 7.326
C(product_id)[10] 7.3350 2.43e-13 3.01e+13 0.000 7.335 7.335
C(product_id)[11] 7.6404 2.43e-13 3.14e+13 0.000 7.640 7.640
C(product_id)[12] 8.3380 2.43e-13 3.43e+13 0.000 8.338 8.338
C(product_id)[13] 7.3665 2.43e-13 3.03e+13 0.000 7.366 7.366
C(product_id)[14] 7.4784 2.43e-13 3.07e+13 0.000 7.478 7.478
C(product_id)[15] 7.5034 2.43e-13 3.08e+13 0.000 7.503 7.503
C(product_id)[16] 8.4451 2.43e-13 3.47e+13 0.000 8.445 8.445
C(product_id)[17] 8.5662 2.43e-13 3.52e+13 0.000 8.566 8.566
C(product_id)[18] 8.1309 2.43e-13 3.34e+13 0.000 8.131 8.131
C(product_id)[19] 7.9042 2.43e-13 3.25e+13 0.000 7.904 7.904
C(product_id)[20] 8.0474 2.43e-13 3.31e+13 0.000 8.047 8.047
C(product_id)[21] 8.2976 2.43e-13 3.41e+13 0.000 8.298 8.298
C(product_id)[22] 7.2276 2.43e-13 2.97e+13 0.000 7.228 7.228
C(product_id)[23] 8.2366 2.43e-13 3.39e+13 0.000 8.237 8.237
C(product_id)[24] 6.9983 2.43e-13 2.88e+13 0.000 6.998 6.998
C(product_id)[25] 7.3282 2.43e-13 3.01e+13 0.000 7.328 7.328
C(product_id)[26] 8.0522 2.43e-13 3.31e+13 0.000 8.052 8.052
C(product_id)[27] 6.8698 2.43e-13 2.82e+13 0.000 6.870 6.870
C(product_id)[28] 7.6842 2.43e-13 3.16e+13 0.000 7.684 7.684
C(product_id)[29] 7.2468 2.43e-13 2.98e+13 0.000 7.247 7.247
C(product_id)[30] 7.5363 2.43e-13 3.1e+13 0.000 7.536 7.536
C(product_id)[31] 5.3634 2.43e-13 2.2e+13 0.000 5.363 5.363
C(product_id)[32] 6.5901 2.43e-13 2.71e+13 0.000 6.590 6.590
C(product_id)[33] 7.3707 2.43e-13 3.03e+13 0.000 7.371 7.371
C(product_id)[34] 8.1125 2.43e-13 3.33e+13 0.000 8.112 8.112
C(product_id)[35] 7.5917 2.43e-13 3.12e+13 0.000 7.592 7.592
C(product_id)[36] 7.6806 2.43e-13 3.16e+13 0.000 7.681 7.681
C(product_id)[37] 8.1022 2.43e-13 3.33e+13 0.000 8.102 8.102
C(product_id)[38] 7.2121 2.43e-13 2.96e+13 0.000 7.212 7.212
C(product_id)[39] 7.8071 2.43e-13 3.21e+13 0.000 7.807 7.807
C(product_id)[40] 7.3393 2.43e-13 3.02e+13 0.000 7.339 7.339
C(product_id)[41] 6.6928 2.43e-13 2.75e+13 0.000 6.693 6.693
C(product_id)[42] 7.5599 2.43e-13 3.11e+13 0.000 7.560 7.560
C(product_id)[43] 7.9975 2.43e-13 3.29e+13 0.000 7.998 7.998
C(product_id)[44] 7.9394 2.43e-13 3.26e+13 0.000 7.939 7.939
C(product_id)[45] 7.6915 2.43e-13 3.16e+13 0.000 7.691 7.691
C(product_id)[46] 8.6240 2.43e-13 3.54e+13 0.000 8.624 8.624
C(product_id)[47] 7.9000 2.43e-13 3.25e+13 0.000 7.900 7.900
C(product_id)[48] 8.1462 2.43e-13 3.35e+13 0.000 8.146 8.146
C(product_id)[49] 8.5314 2.43e-13 3.51e+13 0.000 8.531 8.531
C(product_id)[50] 7.0360 2.43e-13 2.89e+13 0.000 7.036 7.036
C(product_id)[51] 7.1342 2.43e-13 2.93e+13 0.000 7.134 7.134
C(product_id)[52] 8.3714 2.43e-13 3.44e+13 0.000 8.371 8.371
C(product_id)[53] 7.2526 2.43e-13 2.98e+13 0.000 7.253 7.253
C(product_id)[54] 7.6367 2.43e-13 3.14e+13 0.000 7.637 7.637
C(product_id)[55] 8.2654 2.43e-13 3.4e+13 0.000 8.265 8.265
C(product_id)[56] 8.0738 2.43e-13 3.32e+13 0.000 8.074 8.074
C(product_id)[57] 8.4258 2.43e-13 3.46e+13 0.000 8.426 8.426
C(product_id)[58] 8.4041 2.43e-13 3.45e+13 0.000 8.404 8.404
C(product_id)[59] 8.1177 2.43e-13 3.34e+13 0.000 8.118 8.118
C(product_id)[60] 7.2436 2.43e-13 2.98e+13 0.000 7.244 7.244
C(product_id)[61] 8.0709 2.43e-13 3.32e+13 0.000 8.071 8.071
C(product_id)[62] 7.8440 2.43e-13 3.22e+13 0.000 7.844 7.844
C(product_id)[63] 6.9582 2.43e-13 2.86e+13 0.000 6.958 6.958
C(product_id)[64] 8.2601 2.43e-13 3.39e+13 0.000 8.260 8.260
C(product_id)[65] 8.0163 2.43e-13 3.29e+13 0.000 8.016 8.016
C(product_id)[66] 7.7168 2.43e-13 3.17e+13 0.000 7.717 7.717
C(product_id)[67] 7.5826 2.43e-13 3.12e+13 0.000 7.583 7.583
C(product_id)[68] 6.9786 2.43e-13 2.87e+13 0.000 6.979 6.979
C(product_id)[69] 7.8119 2.43e-13 3.21e+13 0.000 7.812 7.812
C(product_id)[70] 7.8770 2.43e-13 3.24e+13 0.000 7.877 7.877
C(product_id)[71] 7.3204 2.43e-13 3.01e+13 0.000 7.320 7.320
C(product_id)[72] 8.1196 2.43e-13 3.34e+13 0.000 8.120 8.120
C(product_id)[73] 7.0714 2.43e-13 2.91e+13 0.000 7.071 7.071
C(product_id)[74] 6.3094 2.43e-13 2.59e+13 0.000 6.309 6.309
C(product_id)[75] 8.2560 2.43e-13 3.39e+13 0.000 8.256 8.256
C(product_id)[76] 7.5130 2.43e-13 3.09e+13 0.000 7.513 7.513
C(product_id)[77] 7.7159 2.43e-13 3.17e+13 0.000 7.716 7.716
C(product_id)[78] 7.2555 2.43e-13 2.98e+13 0.000 7.255 7.255
C(product_id)[79] 6.6701 2.43e-13 2.74e+13 0.000 6.670 6.670
C(product_id)[80] 8.2157 2.43e-13 3.38e+13 0.000 8.216 8.216
C(product_id)[81] 7.3669 2.43e-13 3.03e+13 0.000 7.367 7.367
C(product_id)[82] 8.3464 2.43e-13 3.43e+13 0.000 8.346 8.346
C(product_id)[83] 8.0432 2.43e-13 3.31e+13 0.000 8.043 8.043
C(product_id)[84] 7.6967 2.43e-13 3.16e+13 0.000 7.697 7.697
C(product_id)[85] 7.1042 2.43e-13 2.92e+13 0.000 7.104 7.104
C(product_id)[86] 7.3010 2.43e-13 3e+13 0.000 7.301 7.301
C(product_id)[87] 7.0476 2.43e-13 2.9e+13 0.000 7.048 7.048
C(product_id)[88] 8.4550 2.43e-13 3.47e+13 0.000 8.455 8.455
C(product_id)[89] 7.7943 2.43e-13 3.2e+13 0.000 7.794 7.794
C(product_id)[90] 8.1507 2.43e-13 3.35e+13 0.000 8.151 8.151
C(product_id)[91] 6.8847 2.43e-13 2.83e+13 0.000 6.885 6.885
C(product_id)[92] 7.8002 2.43e-13 3.21e+13 0.000 7.800 7.800
C(product_id)[93] 8.2956 2.43e-13 3.41e+13 0.000 8.296 8.296
C(product_id)[94] 7.5659 2.43e-13 3.11e+13 0.000 7.566 7.566
C(product_id)[95] 8.2305 2.43e-13 3.38e+13 0.000 8.230 8.230
C(product_id)[96] 7.0860 2.43e-13 2.91e+13 0.000 7.086 7.086
C(product_id)[97] 6.8607 2.43e-13 2.82e+13 0.000 6.861 6.861
C(product_id)[98] 7.8494 2.43e-13 3.23e+13 0.000 7.849 7.849
C(product_id)[99] 7.5094 2.43e-13 3.09e+13 0.000 7.509 7.509
C(product_id)[100] 8.0233 2.43e-13 3.3e+13 0.000 8.023 8.023
C(product_id)[101] 6.3417 2.43e-13 2.61e+13 0.000 6.342 6.342
C(product_id)[102] 7.9844 2.43e-13 3.28e+13 0.000 7.984 7.984
C(product_id)[103] 8.6347 2.43e-13 3.55e+13 0.000 8.635 8.635
C(product_id)[104] 7.6907 2.43e-13 3.16e+13 0.000 7.691 7.691
C(product_id)[105] 7.7229 2.43e-13 3.17e+13 0.000 7.723 7.723
C(product_id)[106] 7.7729 2.43e-13 3.19e+13 0.000 7.773 7.773
C(product_id)[107] 6.5686 2.43e-13 2.7e+13 0.000 6.569 6.569
C(product_id)[108] 7.8328 2.43e-13 3.22e+13 0.000 7.833 7.833
C(product_id)[109] 6.7378 2.43e-13 2.77e+13 0.000 6.738 6.738
C(product_id)[110] 7.4703 2.43e-13 3.07e+13 0.000 7.470 7.470
C(product_id)[111] 8.2393 2.43e-13 3.39e+13 0.000 8.239 8.239
C(product_id)[112] 7.3117 2.43e-13 3e+13 0.000 7.312 7.312
C(product_id)[113] 8.0588 2.43e-13 3.31e+13 0.000 8.059 8.059
C(product_id)[114] 8.1906 2.43e-13 3.37e+13 0.000 8.191 8.191
C(product_id)[115] 7.8403 2.43e-13 3.22e+13 0.000 7.840 7.840
C(product_id)[116] 7.0018 2.43e-13 2.88e+13 0.000 7.002 7.002
C(product_id)[117] 7.4384 2.43e-13 3.06e+13 0.000 7.438 7.438
C(product_id)[118] 8.5389 2.43e-13 3.51e+13 0.000 8.539 8.539
C(product_id)[119] 6.3615 2.43e-13 2.61e+13 0.000 6.362 6.362
C(product_id)[120] 7.5006 2.43e-13 3.08e+13 0.000 7.501 7.501
C(product_id)[121] 7.4093 2.43e-13 3.05e+13 0.000 7.409 7.409
C(product_id)[122] 7.7839 2.43e-13 3.2e+13 0.000 7.784 7.784
C(product_id)[123] 8.0607 2.43e-13 3.31e+13 0.000 8.061 8.061
C(product_id)[124] 8.3146 2.43e-13 3.42e+13 0.000 8.315 8.315
C(product_id)[125] 7.6237 2.43e-13 3.13e+13 0.000 7.624 7.624
C(product_id)[126] 8.0643 2.43e-13 3.31e+13 0.000 8.064 8.064
C(product_id)[127] 7.9005 2.43e-13 3.25e+13 0.000 7.901 7.901
C(product_id)[128] 7.9376 2.43e-13 3.26e+13 0.000 7.938 7.938
C(product_id)[129] 7.4227 2.43e-13 3.05e+13 0.000 7.423 7.423
C(product_id)[130] 7.1900 2.43e-13 2.96e+13 0.000 7.190 7.190
C(product_id)[131] 7.1632 2.43e-13 2.94e+13 0.000 7.163 7.163
C(product_id)[132] 7.2598 2.43e-13 2.98e+13 0.000 7.260 7.260
C(product_id)[133] 7.7934 2.43e-13 3.2e+13 0.000 7.793 7.793
C(product_id)[134] 8.0539 2.43e-13 3.31e+13 0.000 8.054 8.054
C(product_id)[135] 7.8714 2.43e-13 3.24e+13 0.000 7.871 7.871
C(product_id)[136] 7.4447 2.43e-13 3.06e+13 0.000 7.445 7.445
C(product_id)[137] 8.3851 2.43e-13 3.45e+13 0.000 8.385 8.385
C(product_id)[138] 8.3578 2.43e-13 3.43e+13 0.000 8.358 8.358
C(product_id)[139] 7.5431 2.43e-13 3.1e+13 0.000 7.543 7.543
C(product_id)[140] 7.2658 2.43e-13 2.99e+13 0.000 7.266 7.266
C(product_id)[141] 7.7417 2.43e-13 3.18e+13 0.000 7.742 7.742
C(product_id)[142] 8.3772 2.43e-13 3.44e+13 0.000 8.377 8.377
C(product_id)[143] 7.8428 2.43e-13 3.22e+13 0.000 7.843 7.843
C(product_id)[144] 7.9400 2.43e-13 3.26e+13 0.000 7.940 7.940
C(product_id)[145] 7.5617 2.43e-13 3.11e+13 0.000 7.562 7.562
C(product_id)[146] 7.9411 2.43e-13 3.26e+13 0.000 7.941 7.941
C(product_id)[147] 8.3474 2.43e-13 3.43e+13 0.000 8.347 8.347
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C(product_id)[835] 7.6185 2.43e-13 3.13e+13 0.000 7.619 7.619
C(product_id)[836] 8.2165 2.43e-13 3.38e+13 0.000 8.216 8.216
C(product_id)[837] 8.1357 2.43e-13 3.34e+13 0.000 8.136 8.136
C(product_id)[838] 7.5576 2.43e-13 3.11e+13 0.000 7.558 7.558
C(product_id)[839] 7.4732 2.43e-13 3.07e+13 0.000 7.473 7.473
C(product_id)[840] 7.7227 2.43e-13 3.17e+13 0.000 7.723 7.723
C(product_id)[841] 7.8439 2.43e-13 3.22e+13 0.000 7.844 7.844
C(product_id)[842] 7.1651 2.43e-13 2.94e+13 0.000 7.165 7.165
C(product_id)[843] 8.0175 2.43e-13 3.3e+13 0.000 8.017 8.017
C(product_id)[844] 7.5971 2.43e-13 3.12e+13 0.000 7.597 7.597
C(product_id)[845] 7.4362 2.43e-13 3.06e+13 0.000 7.436 7.436
C(product_id)[846] 7.4418 2.43e-13 3.06e+13 0.000 7.442 7.442
C(product_id)[847] 7.8481 2.43e-13 3.23e+13 0.000 7.848 7.848
C(product_id)[848] 7.4097 2.43e-13 3.05e+13 0.000 7.410 7.410
C(product_id)[849] 7.7299 2.43e-13 3.18e+13 0.000 7.730 7.730
C(product_id)[850] 8.6768 2.43e-13 3.57e+13 0.000 8.677 8.677
C(product_id)[851] 7.7331 2.43e-13 3.18e+13 0.000 7.733 7.733
C(product_id)[852] 7.5405 2.43e-13 3.1e+13 0.000 7.540 7.540
C(product_id)[853] 6.7190 2.43e-13 2.76e+13 0.000 6.719 6.719
C(product_id)[854] 7.7621 2.43e-13 3.19e+13 0.000 7.762 7.762
C(product_id)[855] 7.0741 2.43e-13 2.91e+13 0.000 7.074 7.074
C(product_id)[856] 7.5084 2.43e-13 3.09e+13 0.000 7.508 7.508
C(product_id)[857] 7.8147 2.43e-13 3.21e+13 0.000 7.815 7.815
C(product_id)[858] 6.7492 2.43e-13 2.77e+13 0.000 6.749 6.749
C(product_id)[859] 7.8557 2.43e-13 3.23e+13 0.000 7.856 7.856
C(product_id)[860] 7.8090 2.43e-13 3.21e+13 0.000 7.809 7.809
C(product_id)[861] 7.0330 2.43e-13 2.89e+13 0.000 7.033 7.033
C(product_id)[862] 7.5131 2.43e-13 3.09e+13 0.000 7.513 7.513
C(product_id)[863] 6.6729 2.43e-13 2.74e+13 0.000 6.673 6.673
C(product_id)[864] 8.3155 2.43e-13 3.42e+13 0.000 8.316 8.316
C(product_id)[865] 7.7751 2.43e-13 3.2e+13 0.000 7.775 7.775
C(product_id)[866] 7.6984 2.43e-13 3.16e+13 0.000 7.698 7.698
C(product_id)[867] 7.6072 2.43e-13 3.13e+13 0.000 7.607 7.607
C(product_id)[868] 8.1659 2.43e-13 3.36e+13 0.000 8.166 8.166
C(product_id)[869] 7.6236 2.43e-13 3.13e+13 0.000 7.624 7.624
C(product_id)[870] 7.8678 2.43e-13 3.23e+13 0.000 7.868 7.868
C(product_id)[871] 8.3019 2.43e-13 3.41e+13 0.000 8.302 8.302
C(product_id)[872] 7.5334 2.43e-13 3.1e+13 0.000 7.533 7.533
C(product_id)[873] 7.0827 2.43e-13 2.91e+13 0.000 7.083 7.083
C(product_id)[874] 7.3253 2.43e-13 3.01e+13 0.000 7.325 7.325
C(product_id)[875] 8.2748 2.43e-13 3.4e+13 0.000 8.275 8.275
C(product_id)[876] 7.7161 2.43e-13 3.17e+13 0.000 7.716 7.716
C(product_id)[877] 7.4036 2.43e-13 3.04e+13 0.000 7.404 7.404
C(product_id)[878] 8.0973 2.43e-13 3.33e+13 0.000 8.097 8.097
C(product_id)[879] 8.1859 2.43e-13 3.36e+13 0.000 8.186 8.186
C(product_id)[880] 7.5622 2.43e-13 3.11e+13 0.000 7.562 7.562
C(product_id)[881] 7.1679 2.43e-13 2.95e+13 0.000 7.168 7.168
C(product_id)[882] 6.8340 2.43e-13 2.81e+13 0.000 6.834 6.834
C(product_id)[883] 6.5450 2.43e-13 2.69e+13 0.000 6.545 6.545
C(product_id)[884] 7.5987 2.43e-13 3.12e+13 0.000 7.599 7.599
C(product_id)[885] 7.1866 2.43e-13 2.95e+13 0.000 7.187 7.187
C(product_id)[886] 8.3377 2.43e-13 3.43e+13 0.000 8.338 8.338
C(product_id)[887] 7.7428 2.43e-13 3.18e+13 0.000 7.743 7.743
C(product_id)[888] 6.4662 2.43e-13 2.66e+13 0.000 6.466 6.466
C(product_id)[889] 8.2355 2.43e-13 3.38e+13 0.000 8.235 8.235
C(product_id)[890] 7.7367 2.43e-13 3.18e+13 0.000 7.737 7.737
C(product_id)[891] 7.9408 2.43e-13 3.26e+13 0.000 7.941 7.941
C(product_id)[892] 7.9979 2.43e-13 3.29e+13 0.000 7.998 7.998
C(product_id)[893] 7.8786 2.43e-13 3.24e+13 0.000 7.879 7.879
C(product_id)[894] 7.9424 2.43e-13 3.26e+13 0.000 7.942 7.942
C(product_id)[895] 7.8528 2.43e-13 3.23e+13 0.000 7.853 7.853
C(product_id)[896] 7.0909 2.43e-13 2.91e+13 0.000 7.091 7.091
C(product_id)[897] 7.9212 2.43e-13 3.26e+13 0.000 7.921 7.921
C(product_id)[898] 8.0982 2.43e-13 3.33e+13 0.000 8.098 8.098
C(product_id)[899] 7.5771 2.43e-13 3.11e+13 0.000 7.577 7.577
C(product_id)[900] 7.9476 2.43e-13 3.27e+13 0.000 7.948 7.948
C(product_id)[901] 7.8981 2.43e-13 3.25e+13 0.000 7.898 7.898
C(product_id)[902] 8.4210 2.43e-13 3.46e+13 0.000 8.421 8.421
C(product_id)[903] 7.5274 2.43e-13 3.09e+13 0.000 7.527 7.527
C(product_id)[904] 7.5131 2.43e-13 3.09e+13 0.000 7.513 7.513
C(product_id)[905] 7.6495 2.43e-13 3.14e+13 0.000 7.650 7.650
C(product_id)[906] 7.7089 2.43e-13 3.17e+13 0.000 7.709 7.709
C(product_id)[907] 6.2037 2.43e-13 2.55e+13 0.000 6.204 6.204
C(product_id)[908] 7.3688 2.43e-13 3.03e+13 0.000 7.369 7.369
C(product_id)[909] 7.8803 2.43e-13 3.24e+13 0.000 7.880 7.880
C(product_id)[910] 7.6853 2.43e-13 3.16e+13 0.000 7.685 7.685
C(product_id)[911] 7.3145 2.43e-13 3.01e+13 0.000 7.315 7.315
C(product_id)[912] 7.3323 2.43e-13 3.01e+13 0.000 7.332 7.332
C(product_id)[913] 7.5273 2.43e-13 3.09e+13 0.000 7.527 7.527
C(product_id)[914] 7.2423 2.43e-13 2.98e+13 0.000 7.242 7.242
C(product_id)[915] 7.5862 2.43e-13 3.12e+13 0.000 7.586 7.586
C(product_id)[916] 7.4525 2.43e-13 3.06e+13 0.000 7.452 7.452
C(product_id)[917] 7.9951 2.43e-13 3.29e+13 0.000 7.995 7.995
C(product_id)[918] 8.2709 2.43e-13 3.4e+13 0.000 8.271 8.271
C(product_id)[919] 8.0667 2.43e-13 3.32e+13 0.000 8.067 8.067
C(product_id)[920] 7.8824 2.43e-13 3.24e+13 0.000 7.882 7.882
C(product_id)[921] 7.2832 2.43e-13 2.99e+13 0.000 7.283 7.283
C(product_id)[922] 8.1233 2.43e-13 3.34e+13 0.000 8.123 8.123
C(product_id)[923] 7.5016 2.43e-13 3.08e+13 0.000 7.502 7.502
C(product_id)[924] 7.6659 2.43e-13 3.15e+13 0.000 7.666 7.666
C(product_id)[925] 8.2432 2.43e-13 3.39e+13 0.000 8.243 8.243
C(product_id)[926] 7.8130 2.43e-13 3.21e+13 0.000 7.813 7.813
C(product_id)[927] 7.7818 2.43e-13 3.2e+13 0.000 7.782 7.782
C(product_id)[928] 6.9496 2.43e-13 2.86e+13 0.000 6.950 6.950
C(product_id)[929] 7.9344 2.43e-13 3.26e+13 0.000 7.934 7.934
C(product_id)[930] 7.1360 2.43e-13 2.93e+13 0.000 7.136 7.136
C(product_id)[931] 8.0855 2.43e-13 3.32e+13 0.000 8.086 8.086
C(product_id)[932] 7.6418 2.43e-13 3.14e+13 0.000 7.642 7.642
C(product_id)[933] 6.1305 2.43e-13 2.52e+13 0.000 6.130 6.130
C(product_id)[934] 7.6224 2.43e-13 3.13e+13 0.000 7.622 7.622
C(product_id)[935] 7.6269 2.43e-13 3.13e+13 0.000 7.627 7.627
C(product_id)[936] 7.8604 2.43e-13 3.23e+13 0.000 7.860 7.860
C(product_id)[937] 7.9419 2.43e-13 3.26e+13 0.000 7.942 7.942
C(product_id)[938] 7.5190 2.43e-13 3.09e+13 0.000 7.519 7.519
C(product_id)[939] 8.1416 2.43e-13 3.35e+13 0.000 8.142 8.142
C(product_id)[940] 4.6779 2.43e-13 1.92e+13 0.000 4.678 4.678
C(product_id)[941] 7.3444 2.43e-13 3.02e+13 0.000 7.344 7.344
C(product_id)[942] 7.0908 2.43e-13 2.91e+13 0.000 7.091 7.091
C(product_id)[943] 7.0001 2.43e-13 2.88e+13 0.000 7.000 7.000
C(product_id)[944] 8.1531 2.43e-13 3.35e+13 0.000 8.153 8.153
C(product_id)[945] 8.0317 2.43e-13 3.3e+13 0.000 8.032 8.032
C(product_id)[946] 7.3673 2.43e-13 3.03e+13 0.000 7.367 7.367
C(product_id)[947] 8.1604 2.43e-13 3.35e+13 0.000 8.160 8.160
C(product_id)[948] 7.8096 2.43e-13 3.21e+13 0.000 7.810 7.810
C(product_id)[949] 7.5420 2.43e-13 3.1e+13 0.000 7.542 7.542
C(product_id)[950] 7.5318 2.43e-13 3.1e+13 0.000 7.532 7.532
C(product_id)[951] 7.3143 2.43e-13 3.01e+13 0.000 7.314 7.314
C(product_id)[952] 7.9126 2.43e-13 3.25e+13 0.000 7.913 7.913
C(product_id)[953] 7.9562 2.43e-13 3.27e+13 0.000 7.956 7.956
C(product_id)[954] 7.7603 2.43e-13 3.19e+13 0.000 7.760 7.760
C(product_id)[955] 6.4536 2.43e-13 2.65e+13 0.000 6.454 6.454
C(product_id)[956] 7.8002 2.43e-13 3.21e+13 0.000 7.800 7.800
C(product_id)[957] 7.9831 2.43e-13 3.28e+13 0.000 7.983 7.983
C(product_id)[958] 7.3725 2.43e-13 3.03e+13 0.000 7.372 7.372
C(product_id)[959] 7.7379 2.43e-13 3.18e+13 0.000 7.738 7.738
C(product_id)[960] 6.5568 2.43e-13 2.69e+13 0.000 6.557 6.557
C(product_id)[961] 7.0725 2.43e-13 2.91e+13 0.000 7.073 7.073
C(product_id)[962] 7.5398 2.43e-13 3.1e+13 0.000 7.540 7.540
C(product_id)[963] 7.5215 2.43e-13 3.09e+13 0.000 7.521 7.521
C(product_id)[964] 7.9362 2.43e-13 3.26e+13 0.000 7.936 7.936
C(product_id)[965] 7.0516 2.43e-13 2.9e+13 0.000 7.052 7.052
C(product_id)[966] 7.5784 2.43e-13 3.11e+13 0.000 7.578 7.578
C(product_id)[967] 7.4137 2.43e-13 3.05e+13 0.000 7.414 7.414
C(product_id)[968] 7.9214 2.43e-13 3.26e+13 0.000 7.921 7.921
C(product_id)[969] 8.2158 2.43e-13 3.38e+13 0.000 8.216 8.216
C(product_id)[970] 8.4574 2.43e-13 3.48e+13 0.000 8.457 8.457
C(product_id)[971] 6.7203 2.43e-13 2.76e+13 0.000 6.720 6.720
C(product_id)[972] 7.6817 2.43e-13 3.16e+13 0.000 7.682 7.682
C(product_id)[973] 7.1406 2.43e-13 2.93e+13 0.000 7.141 7.141
C(product_id)[974] 7.6579 2.43e-13 3.15e+13 0.000 7.658 7.658
C(product_id)[975] 7.8297 2.43e-13 3.22e+13 0.000 7.830 7.830
C(product_id)[976] 7.6762 2.43e-13 3.15e+13 0.000 7.676 7.676
C(product_id)[977] 7.6497 2.43e-13 3.14e+13 0.000 7.650 7.650
C(product_id)[978] 8.3501 2.43e-13 3.43e+13 0.000 8.350 8.350
C(product_id)[979] 8.0081 2.43e-13 3.29e+13 0.000 8.008 8.008
C(product_id)[980] 7.5629 2.43e-13 3.11e+13 0.000 7.563 7.563
C(product_id)[981] 7.6653 2.43e-13 3.15e+13 0.000 7.665 7.665
C(product_id)[982] 8.0891 2.43e-13 3.32e+13 0.000 8.089 8.089
C(product_id)[983] 7.9774 2.43e-13 3.28e+13 0.000 7.977 7.977
C(product_id)[984] 7.5364 2.43e-13 3.1e+13 0.000 7.536 7.536
C(product_id)[985] 6.5703 2.43e-13 2.7e+13 0.000 6.570 6.570
C(product_id)[986] 7.0639 2.43e-13 2.9e+13 0.000 7.064 7.064
C(product_id)[987] 7.7705 2.43e-13 3.19e+13 0.000 7.771 7.771
C(product_id)[988] 6.6252 2.43e-13 2.72e+13 0.000 6.625 6.625
C(product_id)[989] 7.6372 2.43e-13 3.14e+13 0.000 7.637 7.637
C(product_id)[990] 7.5456 2.43e-13 3.1e+13 0.000 7.546 7.546
C(product_id)[991] 8.1223 2.43e-13 3.34e+13 0.000 8.122 8.122
C(product_id)[992] 6.9947 2.43e-13 2.87e+13 0.000 6.995 6.995
C(product_id)[993] 6.4298 2.43e-13 2.64e+13 0.000 6.430 6.430
C(product_id)[994] 7.4176 2.43e-13 3.05e+13 0.000 7.418 7.418
C(product_id)[995] 7.9492 2.43e-13 3.27e+13 0.000 7.949 7.949
C(product_id)[996] 8.2117 2.43e-13 3.37e+13 0.000 8.212 8.212
C(product_id)[997] 7.2474 2.43e-13 2.98e+13 0.000 7.247 7.247
C(product_id)[998] 8.0058 2.43e-13 3.29e+13 0.000 8.006 8.006
C(product_id)[999] 7.0997 2.43e-13 2.92e+13 0.000 7.100 7.100
post 3.696e-13 2.18e-14 16.967 0.000 3.27e-13 4.12e-13
treated_post 0.0527 3.07e-14 1.71e+12 0.000 0.053 0.053
==============================================================================
Omnibus: 71.949 Durbin-Watson: 1.955
Prob(Omnibus): 0.000 Jarque-Bera (JB): 202.708
Skew: 0.024 Prob(JB): 9.61e-45
Kurtosis: 4.559 Cond. No. 36.3
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Treated Post Coefficient: 0.0527
Treated Post P-value: 0.0000