We study the trade-off between squared-error accuracy and incentive compatibility in linear regression. Agents report private labels associated with publicly known features and prefer predictions close to their true labels. Ordinary least squares (OLS) need not elicit truthful reports. For regression with $d$ parameters, we design a deterministic group-strategyproof mechanism achieving a $(d+1)$-approximation to the least-squares optimum and prove optimality even among universally strategyproof randomized mechanisms, answering an open question of Chen et al. (EC 2018). Relaxing universal strategyproofness to strategyproofness in expectation reveals a sharp separation: squared individual loss retains the factor $d+1$, while absolute individual loss admits the tight ratio $2-1/(\lceil d/2\rceil+1)$.

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