## 1 Introduction  
Droughs, one of the more common and frequent meteorological disasters, are characterized by their widespread impact and long duration. Many studies have shown that the ongoing global warming process is one of the triggers for more drought in mid-latitudes [1,2]. When drought occurs, water stress initially causes the changes in plant hydrodynamics and then leads to disorders of plant metabolism and slowing of growth processes. Ecosystems are composed of plant communities that are adapted to current water conditions and can withstand moderate water deficits. If droughts last too long or even extreme droughts occur, the ecosystem carbon fixation will decrease [3; 4], and eventually lead to the death of plants [5; 6; 7] and loss of ecosystem function. As the frequency of drought events increase by year globally, it will have more profound impacts on terrestrial biomes [8]. After a drought event, plants begin to recover within a long or short period depending on the environmental condition and the characteristics of their species, trying to adjust their structural state as much as possible to the level before the damage [9; 10]. Herbaceous plants have been found to be more sensitive to changes in their surroundings than woody plants, mainly because they have shallower root systems and lower hydraulic resistance in the xylem, making grasslands vulnerable to drought stress [11; 12]. Studies have shown that there is a certain lag effect of drought events on plant growth [13; 14], which exhibits significant differences across species and geographic regions [14; 15]. The previous interpretations of the interaction between drought and vegetation usually focus on a single point in time, thus there is still a lack of knowledge of the duration and intensity of the effects on plants after the start of drought [16]. Events such as cumulative water deficits can exacerbate the effects of drought on ecosystems [14; 17]. To clarify the variability of plant response time to drought is essential for effective management and conservation of vegetation and the understanding of mechanisms of climate-ecosystem interactions [10; 18]. On the other hand, how to quantify the strength and duration of vegetation ability to regulate climate deserve deep exploration. The time-lag effect refers to the impacts of an event occurring before a specific point in time on the current vegetation growth, while the cumulative effect includes dynamic changes within a certain period. The vector autoregressive model (VAR) is an econometric model [19] that can be used to predict systems of interrelated variables, analyze the dynamic effects of random disturbances caused by independent variables (e.g., climatic factors) on the dependent variable system (e.g., vegetation growth), and consider time-lag effects [20]. VAR model has been applied to analyze the effects of drought on ecosystem health [21]. To examine the time-lag response effects among drought and ecosystem functional variables at finer scale, the extension of the pixel-based VAR model needs to make attempts. Satellite remote sensing provides vital information for studying large-scale and long-term changes in the relationship between drought and plant physiological status that reported in the previous studies [22; 23; 24; 25]. For instance, [PERSON] et al. [15] correlated the Standardized Precipitation Evapotranspiration Index (SPEI) with indicators of vegetation activity and growth such as the Global Inventory Modeling and Mapping Studies-Normalized Difference Vegetation Index (GIMMS-NDVI) and found that about 70% of vegetation growth was affected by drought on a global scale. [PERSON] et al. [26] indicated that there was a significant positive relationship between moisture conditions and vegetation growth in most regions of China by calculating the correlation coefficients between SPEI and GIMMS-NDVI.