The risk persists into adolescence and there is also an increased risk of depressive symptoms.
"The findings emphasise the importance of health professionals spotting mental health problems in the mother and/or the child as early as possible, for example when the child attends their regular health check-ups at the health clinic in the early years," said Wendy Nilsen, head of the study at the Norwegian Institute of Public Health.
"This gives health professionals a unique opportunity to introduce early preventive measures against the development of mental health problems," said Nilsen.
In addition, the children had a higher risk of reporting depressive symptoms during adolescence.
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The association between maternal and later child problem behaviours was already present when the children were 18 months old.
Disruptive and emotional problems and behaviours in the children were not affected by the mothers' mental health.
The researchers found that there was a tendency for disruptive problem behaviours to be a risk factor for later emotional problems, but not vice versa.
The study was published in the Journal of Developmental and Behavioural Pediatrics.