I'm doing a cross-country panel and wondering about the inclusion of time.
I've seen people put time dummies for each year in the regression and others instead put a single time trend variable. It’s probably well-known to practitioners in the field, but what is the difference and interpretation between them? When should you use a time trend and when time dummies?
Many thanks.
I've also asked myself this question, and this is the way I look at it:
Suppose your regression models are
Time dummies
$y_t =\alpha + X_t\beta +\sum_{j=1}^{T-1}\tau_jT_{j} +e_{it}$
where $\tau_j$ is the coefficent on dummy $T_{j}$, the latter equal to one year $j$, zero elsewhere. For any given year, you can evaluate the function by setting $T_j=1$ for $j=$ the year you evaluate, and zero elsewhere. This gives you:
$y_{t=j}=\alpha + X_j\beta+\tau_j$
Thus, you have a year-specific effect of size $\tau_j$ that affects all your units. I view this approach most appropriate if you suspect that there are specific effects to that year, and wish to model them. E.g. the quality of students in a class for a given year, might exhibit year-specific changes.
Time trend
$y_t =\alpha + X_t\beta +\lambda t +e_{it}$
where $\lambda$ is the coefficient on the time trend $t$ increasing with equal steps, e.g. years. To obtain an intepretable expression, you can take the derivate:
$\frac{\partial y_t}{\partial t}= \lambda$
So moving from one year to another, i.e. increasing $t$ by one unit, yields an effect of $\lambda$ on your outcome variable. Thus, you have a linear trend which can be intepreted as the overall direction your outcomes moves across time. You assume that the effect you estimate is not specific to any given year, but the process which generates the changes extends across years - that's at least how I think about it.
The way I see it, it's more question of what you want to estimate. Year-specific changes or trends (or you might want to compare which of these models is the most appropriate).
Preface: I assume here that modeling time is of secondary concern (i.e., you want to control out the effects of time in order to more accurately capture the primary effects)
It's a matter of parsimony.
If your time trend can be well approximated by a linear trend term or linear plus quadratic trend terms (or possibly something more complicated), you should opt for those in place of time dummies. Assuming many time periods, the simpler linear or quadratic time trend terms will result in more parsimony of the model. But if you have no reason to believe the trend over time is so simple, then dummies are frankly a safer bet if you can afford the complexity of the additional parameters.
One advantage of using a time trend is that you forecast panel-dependent variables for the future by projecting the time trend (linear or non-linear). This will be a problem with time dummies, e.g. to predict your dependent variable in 2024, what time adjustment is to use to forecast your dependent variable. This is especially interesting for the machine learning community who are more interested in prediction.
When 2 time series are purely driven by time only (increasing or decreasing), their correlation or regression is spurious. The reason is this: Both series are increasing or decreasing merely due to time but they may not necessarily co-move i.e. the patterns of movement may not be congruent. This time movement can be captured by time trend variable itself and we don't need any variable to explain the variation in dependent variable apart from time trend. This means the dependent variable is highly correlated with time only than any other variable. In contrast, if the dependent variable movement is more correlated with independent variable movement than the time, there exists a non-spurious and real correlation between2 variables. This comparison of comovement between variables and time can be easily judged by correlations between them. Generally, if correlation between dep var and indep var is higher than that between dep var & time, indep variable is really correlated with dep var without any doubt of spurious relation. If not, then it is time that can take care of movement of dependent variable and independent variable remians useless or insignificant in regression model. Thus, by ingesting a time trend variable, we control for time effect in the model to get the true and non-spurious relationship between dep and indep variables.
