We quantify the bullwhip effect (which measures how the variance of replenishment orders is amplified as the orders move up the supply chain) when demands constitute a first-order auto-regressive random process and lead times constitute a possibly temporally correlated stationary sequence of random variables. We assume future demands are predicted with the minimum mean squared error method and random lead times are estimated using any method. Under these general assumptions we derive a formula for the bullwhip effect measure as the ratio of the replenishment orders variance and demands variance. Using this formula we analyse the impact of auto-correlated demands and auto-correlated lead times on the bullwhip effect. Our investigation of the impact of the lead time auto-correlation on bullwhip appears to be unique in the literature. Our analysis focuses on using the naive forecasting method, the moving average method and the minimum mean squared error method for forecasting the lead times. We show how the bullwhip effect is influenced by demand auto-correlation, lead time auto-correlation, and number of periods in the moving average forecast of the lead times. We reveal that there exists minima and maxima in bullwhip effect as a function of those parameters. For the moving average forecasting method of lead times and their negative auto-correlation we observe an even-odd phenomenon. Our theoretical results are confirmed by Monte Carlo simulation.