Interpretation of coefficients (Log-log/linear)

[Adam Klaas]

Hello Statalist users,

This is my dataset:

* Example generated by -dataex-. For more info, type help dataex
clear
input float log_realHP double(Unemployment_rate CPI_index) float(logrealconsind logReal_income real_interest)
11.066638                  .                100         .         .         .
11.448387                  .              102.4         .         .         .
11.449027                  .           106.5984         .         .         .
11.436362                  . 110.11614719999999         .         .     3.355
11.425336                  . 112.42858629119998         .         .  3.809167
 11.88968                  . 113.89015791298557         .         .  3.126667
11.878828                  . 115.82629059750631         .         .  3.594167
  11.9539 1.1709528024032914 117.10037979407888         .         .  3.216667
12.027326 1.0189597064731957 118.97398587078413         .         .  2.404167
12.061186   1.65181669133134 121.94833551755373         .         . 3.6008334
12.058692 1.7168838356396807 123.41171554376437         .         .  3.459167
 12.03159 1.8364521815338959 125.01606784583329         . 10.228337  2.470833
11.994463 2.3129111842105266 127.89143740628745 11.449235  10.21136 2.2066667
11.930356 2.5779070224387945 131.08872334144462  11.39064 10.195252 2.0708334
  11.8488 2.5307801523129543 134.36594142498072  11.28985 10.225975  3.365833
11.817344 2.3942860219851227 135.70960083923052 11.270553 10.213326    3.5075
11.832868 2.2004945873548722  136.5238584442659  11.32823 10.241876 3.4741666
11.876635 1.9359284664104206  136.9334300195987 11.364115 10.274782 2.0316668
11.936114 1.5915721050958196 138.85049803987306  11.46233 10.283296 1.4991666
12.051608  1.363041781627516  141.2109565065509 11.507878  10.30509     .4025
12.139122                  .  144.8824413757212 11.534215         .     1.435
11.373663                  .                100         .         .         .
  11.8328                  .              102.4         .         .         .
11.792617                  .           106.5984         .         .         .
11.767217                  . 110.11614719999999         .         .     3.355
11.753452                  . 112.42858629119998         .         .  3.809167
12.143667                  . 113.89015791298557         .         .  3.126667
 12.12681                  . 115.82629059750631         .         .  3.594167
12.205215 1.3994621962031346 117.10037979407888         .         .  3.216667
 12.26741 1.2629286880783888 118.97398587078413         .         .  2.404167
12.281482 1.9913041609661408 121.94833551755373         .         . 3.6008334
12.280895 1.9178043271348002 123.41171554376437         .         .  3.459167
12.225747 1.9098898465745313 125.01606784583329         . 10.303476  2.470833
12.191173   2.37366700042712 127.89143740628745 11.449235 10.286084 2.2066667
12.096637 2.7690231192558423 131.08872334144462  11.39064  10.26936 2.0708334
 12.01052 2.6149684400360687 134.36594142498072  11.28985 10.301238  3.365833
11.996036 2.3745448788982113 135.70960083923052 11.270553 10.293784    3.5075
12.025772 2.2898609680403648  136.5238584442659  11.32823 10.326927 3.4741666
12.065701  2.007377765032522  136.9334300195987 11.364115  10.35463 2.0316668
12.124705 1.7314613663629586 138.85049803987306  11.46233 10.368253 1.4991666
 12.19029 1.5859610214620499  141.2109565065509 11.507878 10.389135     .4025
12.223978                  .  144.8824413757212 11.534215         .     1.435
11.289782                  .                100         .         .         .
11.818513                  .              102.4         .         .         .
  11.7855                  .           106.5984         .         .         .
11.753032                  . 110.11614719999999         .         .     3.355
11.739367                  . 112.42858629119998         .         .  3.809167
12.134277                  . 113.89015791298557         .         .  3.126667
 12.11742                  . 115.82629059750631         .         .  3.594167
12.226357               1.33 117.10037979407888         .         .  3.216667
 12.26741               1.05 118.97398587078413         .         .  2.404167
12.273849 1.6038806086922506 121.94833551755373         .         . 3.6008334
12.269553 1.6676362226100447 123.41171554376437         .         .  3.459167
12.249003 1.7095345403683269 125.01606784583329         .  10.33514  2.470833
12.199078 2.3100113542930973 127.89143740628745 11.449235 10.327867 2.2066667
 12.11769 2.4056473557597506 131.08872334144462  11.39064 10.308275 2.0708334
12.024005 2.3806689679800024 134.36594142498072  11.28985 10.347644  3.365833
11.982306 2.2976666798716474 135.70960083923052 11.270553 10.330508    3.5075
11.976323 2.2146642410820214  136.5238584442659  11.32823 10.366885 3.4741666
 12.01838 2.0086648286727056  136.9334300195987 11.364115 10.391165 2.0316668
 12.01325  1.615063420783109 138.85049803987306  11.46233 10.399435 1.4991666
12.068002  1.218314010611122  141.2109565065509 11.507878 10.417065     .4025
12.101523                  .  144.8824413757212 11.534215         .     1.435
11.552146                  .                100         .         .         .
12.104395                  .              102.4         .         .         .
12.069604                  .           106.5984         .         .         .
12.047832                  . 110.11614719999999         .         .     3.355
12.032354                  . 112.42858629119998         .         .  3.809167
12.387163                  . 113.89015791298557         .         .  3.126667
12.373962                  . 115.82629059750631         .         .  3.594167
  12.3918  .7958766012279239 117.10037979407888         .         .  3.216667
12.434464  .6784260515603799 118.97398587078413         .         .  2.404167
12.436175 1.1649021824785006 121.94833551755373         .         . 3.6008334
12.440403 1.1593626817715061 123.41171554376437         .         .  3.459167
12.424276 1.2185482808589119 125.01606784583329         . 10.482217  2.470833
 12.37217 1.4044026389705402 127.89143740628745 11.449235  10.48165 2.2066667
12.310374 2.0481810201892126 131.08872334144462  11.39064 10.450358 2.0708334
 12.23276 2.0915789812401457 134.36594142498072  11.28985  10.50418  3.365833
12.226426 1.9489440557206301 135.70960083923052 11.270553  10.46103    3.5075
12.241873  1.880597966404233  136.5238584442659  11.32823 10.506447 3.4741666
 12.27701  1.492110860662243  136.9334300195987 11.364115 10.546596 2.0316668
12.296556 1.2607160867372669 138.85049803987306  11.46233  10.54604 1.4991666
12.403312 1.1927555792711635  141.2109565065509 11.507878  10.57182     .4025
12.442365                  .  144.8824413757212 11.534215         .     1.435
11.127263                  .                100         .         .         .
11.796694                  .              102.4         .         .         .
11.756512                  .           106.5984         .         .         .
11.724045                  . 110.11614719999999         .         .     3.355
11.710588                  . 112.42858629119998         .         .  3.809167
 12.07601                  . 113.89015791298557         .         .  3.126667
12.059152                  . 115.82629059750631         .         .  3.594167
12.111186 1.0519442832269297 117.10037979407888         .         .  3.216667
12.145667  .9472802127737094 118.97398587078413         .         .  2.404167
 12.14305 1.6363636363636365 121.94833551755373         .         . 3.6008334
12.139817 1.4213345967418638 123.41171554376437         .         .  3.459167
 12.10502  1.501556491485076 125.01606784583329         . 10.279052  2.470833
 12.04164  1.957019422494646 127.89143740628745 11.449235 10.261792 2.2066667
11.984159  2.443268056121127 131.08872334144462  11.39064 10.239828 2.0708334
11.880217 2.4531668153434434 134.36594142498072  11.28985  10.27592  3.365833
11.833517 1.9693816884661117 135.70960083923052 11.270553 10.250465    3.5075
end

My question is how do I interpret the following regression:

-----------------------------------------------------------------------------------
                  |               Robust
       log_realHP |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
   logReal_income |    1.69775   .6003548     2.83   0.007     .4834174    2.912082
Unemployment_rate |   .0564982   .0183193     3.08   0.004     .0194439    .0935524
        CPI_index |  -.0170389   .0018809    -9.06   0.000    -.0208435   -.0132343
    real_interest |  -.0493594   .0135287    -3.65   0.001    -.0767238   -.0219949
   logrealconsind |   .5121079   .0954431     5.37   0.000     .3190561    .7051598
                  |
             Year |
            2013  |   .0292081   .0227494     1.28   0.207    -.0168068     .075223
            2014  |   .0832902    .009536     8.73   0.000     .0640019    .1025785
            2015  |   .1190992   .0165152     7.21   0.000      .085694    .1525043
            2016  |   .0719323   .0104104     6.91   0.000     .0508753    .0929892
                  |
            _cons |  -9.113447   6.896683    -1.32   0.194    -23.06331    4.836411
------------------+----------------------------------------------------------------
          sigma_u |  .10920836
          sigma_e |  .03188331
              rho |  .92145988   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------

I thought it would be like this:
When logReal_income increases by 1 percent, log_realHP(houseprices) increases by 1.69 percent.
When Unemployment rate increases by 1 percent point, log_realHP(houseprices) increases by 0.0565 percent points.
When CPI index increases by 1 percent point, log_realHP(houseprices) decreases by 0.017
When real_interest increases by 1 percent point, log_realHP(houseprices) decreases by 0.049
When logrealconsind(construction costs) increases by 1 percent, log_realHP(houseprices) increases by 0.512 percent.

Wondering if this is the correct interpretation?

Thank you very much everyone!

[Clyde Schechter]

First, let me point out that you are using observational data, and you do not appear to have a study design that, with high confidence, identifies causal effects. So you should not use causal language for any of these. You should speak of these only as one thing being associated with another.

Next, some of the quantitative statements are wrong.

When logReal_income increases by 1 percent, log_realHP(houseprices) increases by 1.69 percent.

Incorrect. In fact, what happens to log_realHP when logReal_income increases by 1% cannot be determined from this kind of model, and it would depend on the starting value of log_Real_income in any model that could estimate that. I think what you mean to say here is that when Real income increases by 1 percent, the associated increase in real HP is 1.69 percent. This is based on a rule of thumb converting coefficients of log-log regressions into elasticities. That rule of thumb is actually an approximation, and if you did the exact calculation here, you would get 1.70 percent. But good enough.

When Unemployment rate increases by 1 percent point, log_realHP(houseprices) increases by 0.0565 percent points.

This one is not quite correct either. When unemployment rate increases by 1 percentage point, the associated difference in log_realHP is 0.0565 (a dimensionless number, and not a percentage, nor percentage points). Then, if you want to go a step farther, if log_realHP is increasing by 0.0565, realHP itself is multiplied by exp(0.0565) = 1.058 (to three decimal places), which could also be said as real HP (not log_realHP) show an associated increase of 5.8 percent.

When CPI index increases by 1 percent point, log_realHP(houseprices) decreases by 0.017

Correct, except for the use of causal-sounding language.

When real_interest increases by 1 percent point, log_realHP(houseprices) decreases by 0.049

Correct, except for the use of causal-sounding language.

When logrealconsind(construction costs) increases by 1 percent, log_realHP(houseprices) increases by 0.512 percent.

This is wrong in the same way the first one was wrong. When realconsind (not its logarithm) increases by 1 percent, the associated difference in real HP (not its logarithm) is 0.512 percent. (Actually, again, this is based on an approximate rule of thumb, but the exact value is close enough to this for practical purposes.)

[Adam Klaas]

First, let me point out that you are using observational data, and you do not appear to have a study design that, with high confidence, identifies causal effects. So you should not use causal language for any of these. You should speak of these only as one thing being associated with another.

Next, some of the quantitative statements are wrong.


Incorrect. In fact, what happens to log_realHP when logReal_income increases by 1% cannot be determined from this kind of model, and it would depend on the starting value of log_Real_income in any model that could estimate that. I think what you mean to say here is that when Real income increases by 1 percent, the associated increase in real HP is 1.69 percent. This is based on a rule of thumb converting coefficients of log-log regressions into elasticities. That rule of thumb is actually an approximation, and if you did the exact calculation here, you would get 1.70 percent. But good enough.


This one is not quite correct either. When unemployment rate increases by 1 percentage point, the associated difference in log_realHP is 0.0565 (a dimensionless number, and not a percentage, nor percentage points). Then, if you want to go a step farther, if log_realHP is increasing by 0.0565, realHP itself is multiplied by exp(0.0565) = 1.058 (to three decimal places), which could also be said as real HP (not log_realHP) show an associated increase of 5.8 percent.


Correct, except for the use of causal-sounding language.


Correct, except for the use of causal-sounding language.


This is wrong in the same way the first one was wrong. When realconsind (not its logarithm) increases by 1 percent, the associated difference in real HP (not its logarithm) is 0.512 percent. (Actually, again, this is based on an approximate rule of thumb, but the exact value is close enough to this for practical purposes.)

Thanks a lot it makes a lot of sense I will be making these changes. I have one more question, if I have a dummy variable for time and for a specific time for example; Year 2013 the dummy has a coefficient of -0.011. How would I interpret it in that case? A one year increase in year is associated with a decrease of -0.011 on the log house price?

[Clyde Schechter]

if I have a dummy variable for time and for a specific time for example; Year 2013 the dummy has a coefficient of -0.011. How would I interpret it in that case? A one year increase in year is associated with a decrease of -0.011 on the log house price?

No, not quite. You have to think about what a unit change in the year 2013 indicator ("dummy") means. When that indicator changes from 0 to 1 it changes from not 2013 to 2013. So it's not necessarily a one year increase. It's not even necessarily an increase at all. It could reflect the difference between 2006 and 2013, or 2020 and 2013, or anything but 2013 and 2013. So the interpretation would be that the expected value of log house price is 0.011 lower in 2013 than in other years. If you wanted to put it in the (unlogged) house price metric, exp(-0.011) is 0.989. So you could also say it as expected housing prices are 1.1 percent lower in 2013 than in other years.

Added: I'm interpreting your remark as meaning that year2013 is an isolated indicator for that year, and not just one of a series of year indicators. If it's one of a series of year indicators, then it reflects the difference in log real housing price between year 2013 and whatever year(s) is(are) not represented in the series of year indicators.