# Interpretation in log linear regressions with coefficients bigger than 1

Just to clarify something regarding coefficients bigger than 1 in log-linear regressions.

If we have this regression, how would we go about to interpret the 1.12?
D1 is dummy variable for having a car D2 is dummy variable for being us resident lninv is the ln of the investment

Linear regression                                  Number of obs =     580
F(  4,   575) =   45.58
Prob > F      =  0.0000
R-squared     =  0.1983
Root MSE      =  1.2828
Robust
lninv   |    Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------+---------------------------------------------------------------
age     |  .1376738   .0221547     6.21   0.000     .0941598    .1811878
1.car   |  .5398736   .1179852     4.58   0.000     .3081391     .771608
1.USA   |  1.124221   .1959553     5.74   0.000     .7393456    1.509097
car#USA |
1 1  |  -.0202977  .2457097    -0.08   0.934    -.5028958    .4623003
_cons   |  -.0585373  .1228857    -0.48   0.634    -.2998969    .1828223


so we have lninv = age + d1(car) + d2(usa) + d1.d2

The trouble is the interpretation of the USA.

(1) Is it correct to say that all things equal being a US resident increases the investment by 112%?
Also the interaction dummy is not significant. (2) Is it ok to check for interaction between two binary variables?

1) If the interaction is not significant, then re-run it without the interaction. It's incorrect to pick and choose the coefficients present in the model based on their significance level because it violates the basic operation of algebra.

Suppose the coefficient for USA remains 1.12 in the revised model (aka, the one with the interaction term removed), then given all else equal:

$\hat{lninv_{USA=1}}-\hat{lninv_{USA=0}} = 1.12$

$ln(\hat{inv_{USA=1}}/\hat{inv_{USA=0}}) = 1.12$

$\hat{inv_{USA=1}}/\hat{inv_{USA=0}} = exp^{1.12} = 3.06$

The ratio is actually close to 300%, not 112%. The "multiplying the coefficient by 100 and interpret it as percent change" method is only applicable to coefficient within the range of about -.08 to .08. Beyond this range, the original and the exponent of it can start to deviate considerably.

2) Yes, it is okay to check for interaction between two binary variables, if you have a strong reason to believe such interaction can exist.

Q: Can you clarify what you meant by " It's incorrect to pick and choose the coefficients present in the model based on their significance level because it violates the basic operation of algebra"; and then you asked me to remove it? Without the interaction I get a coefficient of 1.11. Does it make sense also to eliminate the intercept since it makes no economic sense and is not significant?

A: If the regression is

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1*x_2$

all predictors are binary and $\beta_3$ is not statistically significant, when we interpret the effect of $x_1$, we cannot just take $\beta_1$ (like what you did in the original question). When $x_1 = 1$ and $x_2 = 1$, $\beta_3$ will also be added to $\beta_1$, even the extra effect is tiny. We cannot just ignore it.

A better way is to refit the model:

$y = \gamma_0 + \gamma_1 x_1 + \gamma_2 x_2$

and interpret $\gamma_1$. What your original question alluding to is interpreting $\beta_1$ as if it is $\gamma_1$, that's where the algebraic operation had gone wrong.

Q: Now regarding the percentage thing if I take exp 0.07 for example; i get 1.07 which means a 7% percent increase. If we take exp 1.12 we get 3.06 as you rightly said.. do you have to deduct 1 as in the case of the 1.07 to show the increase.. i.e. is the increase only 200%?

A: This is the classical confusion on reporting percentage as a factor of change or as absolute increase. My own approach is to

1. keep the reporting format consistent within one document, and

2. provide the actual statistics that contribute to the ratio/percent increase so that viewers will not be misguided. E.g. Compared to country A citizens, country B citizens invested 3 times as much in 2011-2012 (3.33 units for country A versus 9.99 units.)

For your 3.06, yes, I would agree that "a 200% increase" interpretation is correct. "A 3 times change" or "a 300% change" are also acceptable.

• Just want to add that the "quick" interpretation that the OP posted, of $\beta$ = % change in $Y$ is an approximation from $ln{a \over b} \approx \% \Delta$, which is only good when the difference is small. – Affine Jun 11 '13 at 16:13
• Aren't you ignoring the interaction between car and USA? – Dimitriy V. Masterov Jun 11 '13 at 16:56
• @DimitriyV.Masterov That's why I said "Suppose the coefficient for USA remains 1.12 in the revised model." I will edit it to make it more explicit. Thanks! – Penguin_Knight Jun 11 '13 at 17:37
• That seems like a different question to me than the OP is asking. – Dimitriy V. Masterov Jun 11 '13 at 17:41
• @DimitriyV.Masterov Then answer it the way you see fit. This is what this forum is meant to be. And also, feel free to edit my answer. I don't mind. – Penguin_Knight Jun 11 '13 at 17:42