Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random vectors with values in the Euclidean plane. We prove that the limiting distribution for a properly normalized quadratic functional $$ \int(r(x)-\hat r_n(x))^2\hat h_n^2(x)p(x)\,dx $$is normal$(0,\sigma^2)$, where$r_n(x)$is an estimate of the regression line$r(x)$of the form (1). We obtain also the limiting distribution in case of a sequence of «local» alternatives of the form (7). Finally, for the rate of convergence of moments, we have$$ |\nu_{n,2k}-\nu_{2k}|\le c_1(k,\sigma)n^{-\frac{1}{2}+\delta},\qquad |\nu_{n,2k+1}|\le c_2(k,\sigma)n^{-\frac{1}{4}+\delta}, $$where$c_1(k,\sigma)$and$c_2(k,\sigma)$are some constants which depend on the order$k$of the moment and variance$\sigma^2$.