How is it possible to obtain a good linear regression model when there is no substantial correlation between the output and the predictors? I have trained a linear regression model, using a set of variables/features. And the model has a good performance. However, I have realized that there is no variable with a good correlation with the predicted variable. How is it possible?
 A: A pair of variables may show high partial correlation (the correlation accounting for the impact of other variables) but low - or even zero - marginal correlation (pairwise correlation).
Which means that pairwise correlation between a response, y and some predictor, x may be of little value in identifying suitable variables with (linear) "predictive" value among a collection of other variables.
Consider the following data:
   y  x
1  6  6
2 12 12
3 18 18
4 24 24
5  1 42
6  7 48
7 13 54
8 19 60

The correlation between y and x is $0$. If I draw the least squares line, it's perfectly horizontal and the $R^2$ is naturally going to be $0$.
But when you add a new variable g, which indicates which of two groups the observations came from, x becomes extremely informative:
   y  x g
1  6  6 0
2 12 12 0
3 18 18 0
4 24 24 0
5  1 42 1
6  7 48 1
7 13 54 1
8 19 60 1

The $R^2$ of a linear regression model with both the x and g variables in it will be 1.

It's possible for this sort of thing to happen with every one of the variables in the model - that all have small pairwise correlation with the response, yet the model with them all in there is very good at predicting the response.
Additional reading:
https://en.wikipedia.org/wiki/Omitted-variable_bias
https://en.wikipedia.org/wiki/Simpson%27s_paradox
A: I assume you are training a multiple regression model, in which you have multiple independent variables $X_1$, $X_2$, ..., regressed on Y. The simple answer here is a pairwise correlation is like running an underspecified regression model. As such, you omitted important variables. 
More specifically, when you state "there is no variable with a good correlation with the predicted variable", it sounds like you are checking the pairwise correlation between each independent variable with the dependent variable, Y. This is possible when $X_2$ brings in important, new information and helps clear up the confounding between $X_1$ and Y. With that confounding, though, we may not see a linear pair-wise correlation between $X_1$ and Y. You may also want to check the relationship between partial correlation $\rho_{x_{1},y|x_{2}}$ and multiple regression $y=\beta_1X_1 +\beta_2X_2 + \epsilon$. Multiple regression have a more close relationship with partial correlation than pairwise correlation, $\rho_{x_{1},y}$.
A: In vector terms, if you have a set of vectors $X$ and another vector y, then if y is orthogonal (zero correlation) to every vector in $X$, then it will also be orthogonal to any linear combination of vectors from $X$. However, if the vectors in $X$ have large uncorrelated components, and small correlated components, and the uncorrelated components are linearly dependents, then y can be correlated to a linear combination of $X$. That is, if $X={x_1,x_2 ...}$ and we take $o_i$ = component of x_i orthogonal to y, $p_i$ =  component of x_i parallel to y, then if there exists $c_i$ such that $\sum c_io_i =0$, then  $\sum c_ix_i$ will be parallel to y (i.e., a perfect predictor). If  $\sum c_io_i =0$ is small, then $\sum c_ix_i$ will be a good predictor. So suppose we have $X_1$ and $X_2$ ~ N(0,1) and $E$ ~ N(0,100). Now we create new columns  $X'_1$ and  $X'_2$. For each row, we take a random sample from $E$, add that number to $X_1$ to get $X'_1$, and subtract it from $X_2$ to get $X'_2$. Since each row has the same sample of $E$ being added and subtracted, the $X'_1$ and $X'_2$ columns will be perfect predictors of $Y$, even though each one has just a tiny correlation with $Y$ individually.
