This paper proposes a novel Stackelberg game approach for activating demand response (DR) program in a residential area aiming at addressing the mismatch between the demand and renewable energy generation. In this study, two major players, namely the aggregator as a leader and the consumers as followers, are considered. The aggregator, which owns a wind farm and also receives power from the independent system operator (ISO), strives to obtain the maximum matching between the consumers' demand and forecasted wind power by incentivizing consumers to adjust their load through offering a bonus to them. On the other hand, consumers change their load profiles for obtaining the highest amount of bonuses. Each consumer has two kinds of loads including critical loads, which must be maintained under any circumstances, and the flexible loads, e.g., heating, ventilation, and air conditioning (HVAC) system, which can be regulated for DR purposes. In order to consider the uncertainty associated with the wind generation and the demands, a scenario-based stochastic programming model has been adopted in this work. Results show the effectiveness of the Stackelberg game model used for interaction between the aggregator and consumers, and the best response that can be served to both of them.