Variable selection is not compulsory. The idea that you have to throw away variables is wrong. Actually, unless there are strong reasons to throw away variables, don't do it! Use the full model and use the p-values to guide interpretation rather than throwing away information. An insignificant p-value doesn't mean that a variable should be removed, it only means that the data don't give you clear evidence that the coefficient is nonzero.

The Lasso mitigates some of the issues of doing variable selection by p-values, but as long as you don't feel the need to remove information, there's no need to do it, neither by Lasso.

Good reasons for selecting variables are:

1. The number of observations is critically low for the number of variables you have.

2. There are strong dependences between your variables ($X^TX$ is close to singular) and certain information is representated by several variables

3. The model is used to predict future observations and you are happy to get rid of some variables because it may be costly to observe them in the future.

4. Using cross-validation and the like you find that a model with fewer variables predicts better (if this happens, mostly one of 1 and 2 is also in place, but there are some further situations in which the original set of variables contains a lot of noise).

Variable selection is not required for interpretation! Note that even if there is no evidence (high p-value) that a certain variable has a nonzero coefficient, this variable may still improve the predictive power of the model. Removing the variable will set its coefficient to zero - don't forget that the estimator for it's coefficient in the full model is the "best guess" of the coefficient value that you have, significant or not, and therefore also better than zero in the sense of least squares.

By the way, regarding Lasso vs. variable selection by p-values (backward/forward/stepwise selection - never throw away all variables with large p-values in one go anyway): There are well known issues with variable selection by p-values and Lasso is often better, but not always. I've seen a good number of examples in which a model selected by backward or forward selection did better predicting on independent data than the Lasso. If you have enough observations to properly assess prediction quality using cross-validation and the like, you can compare different approaches and pick the best rather than always using Lasso.

Variable selection is not compulsory. The idea that you have to throw away variables is wrong. Actually, unless there are strong reasons to throw away variables, don't do it! Use the full model and use the p-values to guide interpretation rather than throwing away information. An insignificant p-value doesn't mean that a variable should be removed, it only means that the data don't give you clear evidence that the coefficient is nonzero.

The Lasso mitigates some of the issues of doing variable selection by p-values, but as long as you don't feel the need to remove information, there's no need to do it, neither by Lasso.

Good reasons for selecting variables are:

1. The number of observations is critically low for the number of variables you have.

2. There are strong dependencies between your variables ($X^TX$ is close to singular) and certain information is represented by several variables

3. The model is used to predict future observations and you are happy to get rid of some variables because it may be costly to observe them in the future.

4. Using cross-validation and the like you find that a model with fewer variables predicts better (if this happens, mostly one of 1 and 2 is also in place, but there are some further situations in which the original set of variables contains a lot of noise).

Variable selection is not required for interpretation! Note that even if there is no evidence (high p-value) that a certain variable has a nonzero coefficient, this variable may still improve the predictive power of the model. Removing the variable will set its coefficient to zero - don't forget that the estimator for it's coefficient in the full model is the "best guess" of the coefficient value that you have, significant or not, and therefore also better than zero in the sense of least squares.

By the way, regarding Lasso vs. variable selection by p-values (backward/forward/stepwise selection - never throw away all variables with large p-values in one go anyway): There are well known issues with variable selection by p-values and Lasso is often better, but not always. I've seen a good number of examples in which a model selected by backward or forward selection did better predicting on independent data than the Lasso. If you have enough observations to properly assess prediction quality using cross-validation and the like, you can compare different approaches and pick the best rather than always using Lasso.