# Testing if a regression coeffiecient is larger than another coeffeiceint [duplicate]

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
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
=============================================================================================================