Testing predictor significance with ultra high dimensional multivariate responses

We consider here the problem of testing the effect of a subset of predictors for a regression model with predictor dimension fixed but ultra high dimensional responses. Because the response dimension is ultra high, the classical method of likelihood ratio test is no longer applicable. To solve the problem, we propose a novel solution, which decomposes the original problem into many testing problems with univariate responses. Subsequently, the usual residual sum of squares (RSS) type test statistics can be obtained. Those statistics are then integrated together across different responses to form an overall and powerful test statistic. Under the null hypothesis, the resulting test statistic is asymptotically standard normal after some appropriate standardization. Numerical studies are presented to demonstrate the finite sample performance of the test statistic and a real example about paid search advertising is analyzed for illustration purpose.