Interpreting Different Regression Models
A guide on how to interpret level-level models, level-log models, log-level models, and log-log models with examples using R.
If you have trouble understanding regression models where one or both of the dependent and independent variables are in logarithmic form, or it is your first time learning regression analysis, or you read research papers in the social sciences but do not understand well how researchers explain nonlinear models in their studies, this article is for you.
Level-level Regression Models
Level-level regression models are linear models where the dependent and independent variables are both in their level forms. A simple linear regression equation that has only one independent variable is expressed as:
Y = β0+β1X + e . . . (1.0)
Where Y is the explained variable, β0 is the intercept (sometimes called the constant term), β1 is the slope coefficient, X is the explanatory variable, and e denotes the error or the disturbance term — which represents unobserved factors other than X that affect Y. The slope parameters are the depth of interest in regression analyses. The intercept parameter β0 is not often central to the analysis.
In order to interpret the results of the regression, I assumed here that the Gauss Markov conditions hold.
- Example
I am using the WAGE2 data, a dataset that comes with the Introductory Econometrics: A Modern Approach by Jeffrey M. Wooldridge. The Wooldridge package available in R has all the datasets from the book. But first, you need to install the package and load it, then call any data you want to use for analysis. For newbies, the first line of the code below installs the Wooldridge package, the second loads it, and the third line loads supporting documentation for the dataset in the plots pane (you can read more about the data we are working with this and all the examples that will follow), and the fourth line views the dataset.
install.packages("wooldridge")
library(wooldridge)
?wage2
View(WAGE2)In our examples, we are interested in only two variables; monthly earnings (wage) and the number of years of work experience (exper). A simple linear equation of these two variables is:
wage = β0+ β1exper + e . . . (1.1)
Performing a regression analysis for the above equation to see the linear relationship between wage and exper in the 935 observations of the data as the code below shows.
wage.exper.lm <- lm(wage ~ exper, data = WAGE2)
summary(wage.exper.lm)The output of the above lines of code is the result shown below.
Call:
lm(formula = wage ~ exper, data = WAGE2)Residuals:
Min 1Q Median 3Q Max
-842.43 -289.13 -52.84 201.86 2120.17Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 955.6049 37.4111 25.543 <2e-16 ***
exper 0.2024 3.0261 0.067 0.947
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 404.6 on 933 degrees of freedom
Multiple R-squared: 4.795e-06, Adjusted R-squared: -0.001067
F-statistic: 0.004474 on 1 and 933 DF, p-value: 0.9467
wage= 955.6049 + 0.2024031 exper . . . (1.2)
Equation (1.2) shows the estimated equation for the level-level regression model.
- Interpretation
The intercept of 955.6049 in the above equation is the predicted wage of those who have zero work experience. The slope parameter estimate of 0.2024031 indicates that an additional year of work experience increases monthly earnings by 0.2 dollars. Due to the linear nature of equation (1.2), 0.2 dollars is the increase for either the first year of work experience or the thirtieth year — which may not be practical. A better description of how wage changes with work experience could be a constant percentage increase in earnings with each additional year of work experience, as we will see in the log-level models.
Generally, the slope parameter of level-level regression models tells us the volume of the impact of a one-unit increase in the independent variable on the value of the dependent variable (whether an increase or decrease depends on the sign of the coefficient).
Level-log Regression Models
Level-log regression models are nonlinear models where the explained variable is in its level form, but the explanatory variable or variables are in a logarithmic form. Equation (2.0) shows a level-log regression model of the two variables in our example. Where log denotes a natural logarithm.
wage = β0+ β1log(exper) + e . . . (2.0)
- Example
Run a regression analysis for the level-log equation after transforming exper into a log form, following the two-line code below.
wage.lnexper <- lm(wage ~ log(exper), data = WAGE2)
summary(wage.lnexper)The above code produces the following result.
Call:
lm(formula = wage ~ log(exper), data = WAGE2)Residuals:
Min 1Q Median 3Q Max
-841.64 -289.27 -51.28 199.67 2119.71Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 938.558 66.371 14.141 <2e-16 ***
log(exper) 8.231 27.611 0.298 0.766
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 404.6 on 933 degrees of freedom
Multiple R-squared: 9.523e-05, Adjusted R-squared: -0.0009765
F-statistic: 0.08886 on 1 and 933 DF, p-value: 0.7657
The estimated equation for the level-log model is as shown below.
wage= 938.558+ 8.231log(exper) . . . (2.1)
- Interpretation
The intercept, 938.558, has the same interpretation, the predicted monthly earnings of those with no work experience. However, the slope coefficient indicates that a one-percent increase in the number of years of work experience raises the monthly wage by 0.08231 (β1/100 ) dollars.
Broadly, the slope coefficient of level-log models represents the unit impact on the dependent variable that results from a one-percent increase in the independent variable. The slope parameter in the example above answers the question of how many dollars will earnings increase if the number of years of work experience increases by one percent.
Log-level Regression Models
Log-level regression models are models where the regressand is in a log form, but the regressors are in their level forms. They are the opposite of level-log models. Log-level models roughly give a constant percentage effect which is a better characterization when compared to level-level models. For example, an increase in work experience from five to six years raises monthly earnings, say, 5% (ceteris paribus meaning all other things being equal), and an increase from 10 to 11 years also increases wages by 10%. A bi-variate log-level regression equation is:
log(wage) = β0+ β1exper + e . . . (3.0)
Log-level equations are sometimes referred to as the semi-elasticity of the dependent variable with respect to the independent variable or variables. Semielasticity refers to the change in a function relative to an absolute change in one of its parameters. Mathematically, the semi-elasticity of a function f at point y is expressed as f’(y)/f(y).
- Example
Before running a regression analysis for the log-level model, you need to transform the dependent variable into a log form, as shown in the lines of code below.
Call:
lm(formula = log(wage) ~ exper, data = WAGE2)Residuals:
Min 1Q Median 3Q Max
-2.02899 -0.27363 0.03046 0.27694 1.25415Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.756070 0.038956 173.430 <2e-16 ***
exper 0.001983 0.003151 0.629 0.529
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 0.4213 on 933 degrees of freedom
Multiple R-squared: 0.0004244, Adjusted R-squared: -0.0006469
F-statistic: 0.3961 on 1 and 933 DF, p-value: 0.5292
The estimated equation for the log-level model is expressed in equation (3.1).
log(wage)= 6.756070 + 0.001983exper . . . (3.1)
- Interpretation
The intercept of 6.75607 here, although a different value, has the same meaning. The predicted monthly earnings of non-experienced workers. However, the slope coefficient, 0.00198, means that every additional year of work experience increases the monthly wage by 0.198 percent (0.00198*100 ). The slope parameter of this log-level model shows the percentage impact on monthly earnings that results from a unit change in work experience (measured in years).
In general, slope parameters of log-level models tell us the magnitude of the percentage impact on the dependent variable because of a one-unit change in the independent variable.
Log-log Regression Models
Log-log models are those where both the dependent and independent variables are in natural logarithm forms. Log-log models are sometimes referred to as the elasticity of the dependent variable with respect to one independent variable. Elasticity measures the percentage change or response of Y as a result of a percentage change in X.
A bivariate log-log regression equation for our variables of interest in the WAGE2 data is:
log(wage) = β0+ β1log(exper) + e . . . (4.0)
- Example
The code below converts the two variables into natural logarithms and then runs a regression analysis for the log-log equation.
lnwage.lnexper <- lm(log(wage) ~ log(exper), data = WAGE2)
summary(lnwage.lnexper)The output of the above two-line code above is:
Call:
lm(formula = log(wage) ~ log(exper), data = WAGE2)Residuals:
Min 1Q Median 3Q Max
-2.03085 -0.27405 0.03488 0.27500 1.25217Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.73102 0.06911 97.395 <2e-16 ***
log(exper) 0.02037 0.02875 0.709 0.479
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 0.4213 on 933 degrees of freedom
Multiple R-squared: 0.0005378, Adjusted R-squared: -0.0005334
F-statistic: 0.5021 on 1 and 933 DF, p-value: 0.4788
The estimated equation for the log-log model is:
log(wage)= 6.73102 + 0.02037log(exper) . . . (4.1)
- Interpretation
The intercept, 6.731018 still has the same meaning, the predicted monthly earnings of non-experienced workers. However, the slope coefficient tells us that a one-percent increase in the number of years of work experience raises the monthly wage by 0.02 percent. Notice how log-log models show the percentage impact on monthly earnings for a given percentage change in the number of years of work experience.
As a general rule, the slope coefficient of log-log models is the one-percent impact on the explained variable as a result of a percentage change in the explanatory variable.
Conclusion
To summarize, the slope parameters of the four different regression models are written and interpreted as follows:
- Level-level models are mathematically written as Y = β0+β1X+ e and the coefficient of the slope shows that a unit change in X leads to a unit change in Y.
- Level-log models are expressed as Y=β0+β1log(X)+e. The β1 tells us that a one-percent (β/100) change in X leads to a unit change in Y.
- Log-level models— sometimes called semi-elasticity of Y with respect to X are denoted as log(Y) = β0+βX+ e and the β1 shows the percentage (β*100) change in Y as a result of a unit change in X.
- Log-log models — sometimes called elasticity of Y with respect to X are written as log(Y)= β0+β1log(X) + e. The slope coefficient of these models shows the percentage impact on Y that arises from a percentage change in X.
I hope you found this article helpful. Feel free to check out some of my other related blog posts below.
