If increases in my x var cause increases in my y var does the opposite also hold true?

[John Adler]

I have survey data which records adult income and health across 4 waves. I would like to look at how decreases in income influences respondent health, i.e. as income declines how does health change? In my data income looks like the below:

tab income

 98113.21 |          1        0.01       99.79
   100502.5 |          5        0.05       99.83
   103448.3 |          2        0.02       99.85
   104522.6 |          1        0.01       99.86
   107758.6 |          1        0.01       99.87
     112782 |          1        0.01       99.88
   117735.8 |          1        0.01       99.89
   120689.7 |          1        0.01       99.90
   129310.3 |          1        0.01       99.91
   140939.6 |          1        0.01       99.92
   150753.8 |          1        0.01       99.93
   150862.1 |          2        0.02       99.95
   165829.1 |          1        0.01       99.96
   215517.2 |          2        0.02       99.98
   216867.5 |          1        0.01       99.99
     241206 |          1        0.01      100.00
------------+-----------------------------------
      Total |     10,270      100.00

I want to do a linear probability fixed effects model across all 4 waves, examining how changes in income influence changes in health. However, my understanding is that this model will tell me the change in the outcome for a one-unit increase in my x-variable (here this is income) while what I'd actually like to see if the change in the outcome for a one unit decrease in the x-variable.

In the below toy example x is statistically significant and an additional unit of the x income variable increases the probability of the y-health outcome by 1.1 percentage points. As my results say this, can I say that the opposite also holds true and that one less unit of the x income variable decreases the probability of the y-health outcome by 1.1 percentage points? This would allow me to report the impact of decreases rather than increases in income.

I realize that this is a very simple inquiry but I want to ensure that my approach is correct!

Thanks,

John

 
xtreg y_health_variable x_income_variable control_variable, cluster (region) fe robust
  

Fixed-effects (within) regression               Number of obs      =      1578
Group variable: id                              Number of groups   =       635

R-sq:  within  = 0.0066                         Obs per group: min =         1
       between = 0.0062                                        avg =       2.5
       overall = 0.0047                                        max =         3

                                                F(3,28)            =      4.34
corr(u_i, Xb)  = -0.0538                        Prob > F           =    0.0124

                                                               (Std. Err. adjusted for 29 clusters in current_county_y1)
------------------------------------------------------------------------------------------------------------------------
                                                       |               Robust
                                       binary_health_y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------------------------------------------------+----------------------------------------------------------------
                          x_income_variable|  .0108672   .0043554   2.50   0.019   .0197888   .0019456
                                                        |
                                                 control_variable|   .0074049   .0038814     1.91   0.067    -.0005457    .0153555
                                                 _cons |   .6171357   .0901617     6.84   0.000     .4324479    .8018235
-------------------------------------------------------+----------------------------------------------------------------
                                               sigma_u |  .35499044
                                               sigma_e |  .35438184
                                                   rho |  .50085794   (fraction of variance due to u_i)
------------------------------------------------------------------------------------------------------------------------

[daniel klein]

Well, you are modeling one linear relationship; hence, the effect of a unit change in x is symmetrical. Whether this makes much sense is another question. See Paul Allison's take on this,

Best
Daniel