How to formally test whether a coeffeicent in a regression problem is bigger in magnitude then another coefficient? Here is the model (random effects model):
\begin{align} \mbox{effeciency} = \beta_0 &+ \beta_1 \mbox{square_feet}_{it} + \beta_2 \mbox{wind_speed}_{it} + \beta_3 \mbox{air_temp}_{it} + \beta_4 \mbox{dew_temp}_{it} \\ &+ \beta_5 \mbox{hour}_{it} + \beta_6 \mbox{weekday}_{it} + \beta_7 \mbox{month}_{it} \\ &+ \beta_8 \mbox{edu}_{i} + \beta_9 \mbox{lodge}_{i} + \beta_{10} \mbox{office}_{i} + \beta_{11} \mbox{entertain}_i + \beta_{13} \mbox{site_id}_i + \epsilon_{it} \end{align}
where $i = \mbox{building_id}$
and $t = \mbox{timestamp}$
$\mbox{effeciency} = 1 - \frac{r_{it}}{r_{max} - r_{min}}$
$r_{it} = \frac{\mbox{meter_reading}_{it}}{\mbox{square_feet}_{it}}$
$r_{max} = \max_{i,t} r_{it}$
$r_{min} = \min_{i,t} r_{it}$
For example, when testing whether $\beta_8 > \beta_9$ the test can be formulated as follows:
\begin{aligned} &H_0 : \beta_8 < \beta_9 \\ &H_1 : \beta_8 > \beta_9 \end{aligned}
What will be the test statistic for $H_0$? Also $\beta_8 - \beta_{11}$ are dummy variables with one dummy variable omitted, $\beta_{12} \mbox{public_services}_i $, to avoid collinearity. In addition, how would I test this hypothesis test with the omitted dummy coefficient $\beta_{12}$? All dummy coefficients are the difference from $\beta_{12}$ so would it be $H_0 : \beta_8 < 0$?
Here is the regression summary of the model using linearmodels package in python:
RandomEffects Estimation Summary
================================================================================
Dep. Variable: y R-squared: 0.0957
Estimator: RandomEffects R-squared (Between): 0.0184
No. Observations: 10159489 R-squared (Within): 0.0108
Date: Thu, Mar 19 2020 R-squared (Overall): 0.0154
Time: 01:35:05 Log-likelihood 5.202e+07
Cov. Estimator: Unadjusted
F-statistic: 8.266e+04
Entities: 1199 P-value 0.0000
Avg Obs: 8473.3 Distribution: F(13,10159475)
Min Obs: 479.00
Max Obs: 8782.0 F-statistic (robust): 8559.0
P-value 0.0000
Time periods: 8784 Distribution: F(13,10159475)
Avg Obs: 1156.6
Min Obs: 630.00
Max Obs: 1192.0
Parameter Estimates
=============================================================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
-------------------------------------------------------------------------------------------------------------
const 0.9979 0.0002 5740.4 0.0000 0.9976 0.9983
site_id 1.277e-05 1.252e-05 1.0199 0.3078 -1.177e-05 3.731e-05
square_feet -5.288e-10 5.319e-10 -0.9943 0.3201 -1.571e-09 5.136e-10
air_temperature 5.247e-07 9.145e-08 5.7374 0.0000 3.454e-07 7.039e-07
dew_temperature -9.735e-06 9.482e-08 -102.67 0.0000 -9.921e-06 -9.549e-06
wind_speed 1.354e-06 2.139e-07 6.3320 0.0000 9.352e-07 1.774e-06
hour -1.443e-05 6.775e-08 -213.01 0.0000 -1.456e-05 -1.43e-05
day 1.235e-07 5.147e-08 2.3994 0.0164 2.262e-08 2.244e-07
weekday 4.065e-05 2.271e-07 178.95 0.0000 4.02e-05 4.109e-05
month 3.486e-06 1.376e-07 25.335 0.0000 3.216e-06 3.756e-06
primary_use_Education 0.0003 0.0002 1.6635 0.0962 -5.407e-05 0.0007
primary_use_Entertainment/public assembly 0.0002 0.0002 0.8554 0.3923 -0.0002 0.0006
primary_use_Lodging/residential 0.0010 0.0002 4.1760 0.0000 0.0005 0.0015
primary_use_Office 0.0003 0.0002 1.4861 0.1373 -9.87e-05 0.0007
=============================================================================================================
