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Climate change has been proven to be the ultimate cause of social crisis in pre-industrial Europe at a large scale. However, detailed analyses on climate change and macro-economic cycles in the pre-industrial era remain lacking, especially within different temporal scales. Therefore, fine-grained, paleo-climate, and economic data were employed with statistical methods to quantitatively assess the relations between climate change and agrarian economy in Europe during AD 1500 to 1800. In the study, the Butterworth filter was adopted to filter the data series into a long-term trend (low-frequency) and short-term fluctuations (high-frequency). Granger Causality Analysis was conducted to scrutinize the associations between climate change and macro-economic cycle at different frequency bands. Based on quantitative results, climate change can only show significant effects on the macro-economic cycle within the long-term. In terms of the short-term effects, society can relieve the influences from climate variations by social adaptation methods and self-adjustment mechanism. On a large spatial scale, temperature holds higher importance for the European agrarian economy than precipitation. By examining the supply-demand mechanism in the grain market, population during the study period acted as the producer in the long term, whereas as the consumer in the short term. These findings merely reflect the general interactions between climate change and macro-economic cycles at the large spatial region with a long-term study period. The findings neither illustrate individual incidents that can temporarily distort the agrarian economy nor explain some specific cases. In the study, the scale thinking in the analysis is raised as an essential methodological issue for the first time to interpret the associations between climatic impact and macro-economy in the past agrarian society within different temporal scales.

Only a small number of quantitative studies have investigated the short-and longterm impacts of climate variations on society during Europe's pre-industrial era. Accordingly, there is a lack of research clearly comparing the consequences of climate variation (short-term) and climate change (long-term). This study focuses on the close relationship between climate variations and the dynamics of the agrarian economy in Europe during the period of 1500 to 1800 AD. ARX modeling was applied to analyze the relationship between climate and past agrarian economies, on large spatial and long temporal scales. Both short-and long-term findings are provided, based on statistical analysis, as well as the empirical study of the 17th century economic crisis as a case analysis. The negative climatic variations in the short-term caused grain prices to increase. Grain prices were affected for up to 25 yr by a period of climatic variation due to the price stickiness. The immediate and long-term effects of climate variations over the study period can add up to significantly influence agrarian economies. Climate change occurs when climate variations last for more than 30 yr. The accumulated effect of climate change on the agrarian economy ultimately resulted in an economic crisis in pre-industrial Europe.

a b s t r a c tWe investigated the mechanism of epidemics with the impacts of climate change and socio-economic fluctuations in the Ming and Qing Dynasties in China (AD 1368e1901). Using long-term and highquality datasets, this study is the first quantitative research that verifies the 'climate change / economy / epidemics' mechanism in historical China by statistical methods that include correlation analysis, Granger causality analysis, ARX, and Poisson-ARX modeling. The analysis provides the evidences that climate change could only fundamentally lead to the epidemics spread and occurrence, but the depressed economic well-being is the direct trigger of epidemics spread and occurrence at the national and long term scale in historical China. Moreover, statistical modeling shows that economic well-being is more important than population pressure in the mechanism of epidemics. However, population pressure remains a key element in determining the social vulnerability of the epidemics occurrence under climate change. Notably, the findings not only support adaptation theories but also enhance our confidence to address climatic shocks if economic buffering capacity can be promoted steadily. The findings can be a basis for scientists and policymakers in addressing global and regional environmental changes.

Summary The paper novelly transforms lack‐of‐fit tests for parametric quantile regression models into checking the equality of two conditional distributions of covariates. Accordingly, by applying some successful two‐sample test statistics in the literature, two tests are constructed to check the lack of fit for low and high dimensional quantile regression models. The low dimensional test works well when the number of covariates is moderate, whereas the high dimensional test can maintain the power when the number of covariates exceeds the sample size. The null distribution of the high dimensional test has an explicit form, and the p‐values or critical values can then be calculated directly. The finite sample performance of the tests proposed is examined by simulation studies, and their usefulness is further illustrated by two real examples.

The relationship between climate change and the macroeconomy in pre-industrial Europe has attracted considerable attention in recent years. This study follows the combined paradigms of evolutionary economics and ecological economics, in which wavelet analysis (spectrum analysis and coherence analysis) is applied as the first attempt to examine the relationship between climate change and the macroeconomic structure in pre-industrial Europe in the frequency domain. Aside from confirming previous results, this study aims to further substantiate the association between climate change and macroeconomy by presenting new evidence obtained from the wavelet analysis. Our spectrum analysis shows a consistent and continuous frequency band of 60–80 years in the temperature, grain yield ratio, grain price, consumer price index, and real wage throughout the study period. Besides, coherence analysis shows that the macroeconomic structure is shaped more by climate change than population change. In addition, temperature is proven as a key climatic factor that influences the macroeconomic structure. The analysis reveals a unique frequency band of about 20 years (15–35 years) in the temperature in AD1600-1700, which could have contributed to the widespread economic crisis in pre-industrial Europe. Our findings may have indications in re-examining the Malthusian theory.

Summary Estimating conditional quantiles of financial time series is essential for risk management and many other financial applications. For time series models with conditional heteroscedasticity, although it is the generalized auto‐regressive conditional heteroscedastic (GARCH) model that has the greatest popularity, quantile regression for this model usually gives rise to non‐smooth non‐convex optimization which may hinder its practical feasibility. The paper proposes an easy‐to‐implement hybrid quantile regression estimation procedure for the GARCH model, where we overcome the intractability due to the square‐root form of the conditional quantile function by a simple transformation. The method takes advantage of the efficiency of the GARCH model in modelling the volatility globally as well as the flexibility of quantile regression in fitting quantiles at a specific level. The asymptotic distribution of the estimator is derived and is approximated by a novel mixed bootstrapping procedure. A portmanteau test is further constructed to check the adequacy of fitted conditional quantiles. The finite sample performance of the method is examined by simulation studies, and its advantages over existing methods are illustrated by an empirical application to value‐at‐risk forecasting.