
Statistical models for identifying timing differences (\'leads\' and \'lags\') in the transmission of fluctuations through urban and regional systems. These models have been widely used for studies of regional cycles of economic activity and for studies of spatial diffusion of epidemics such as influenza or measles (cf. medical geography). Three types of timing differences can be studied:
{img src=show_image.php?name=2022.gif }Â leads/lags between different variables within a region; {img src=show_image.php?name=2022.gif }Â leads/lags between regions or cities; {img src=show_image.php?name=2022.gif }Â leads/lags between a regional series and the national aggregate series.The models require detailed timeseries data for the variables and regions, e.g. unemployment data by month or quarter for Canadian cities.
The leadlag structure can be defined using lagged correlation (sometimes called crosscorrelation). To find the leadlag between employment cycles in two regions X and Y, the time series for region X is first correlated with that for region Y to give the usual correlation coefficient r. This is then repeated, but with the Y series lagged by one period, i.e. Xt is related to Yt for t= 2,3,â€¦ T to give r1. This is done for several lags, generating r2, r3, etc., and then for Yt related to X lagged by several periods to give r+1, r+2, r+3, etc. The set of crosscorrelations r+k to rk is then examined to find the highest correlation, so identifying the \'bestfit\' leadlag. It is sometimes appropriate then to use such a leadlag in a lagged regression model, e.g. when examining local fluctuations in response to national economic cycles. Here Yt might be local unemployment and Xt the national unemployment series. If k is the bestfit lag, the regression has the form: Yt = bXtk + et and b measures the cyclical sensitivity of the local economy. It is increasingly recognized, however, that such responses are gradual or distributed, and that accurate modelling requires the incorporation of several lags in the forms of a \'distributed lag\' model. spectral analysis is also used to examine leadlags between regions for different types of cycles.
A drawback of leadlag models is that, like most statistical models, they assume that the relationship is constant across the whole time period under study, e.g. over 12 years and four economic cycles. But each cycle may have very different origins and causes, generating different patterns of leads and lags: downturns may have different timings across space to those of the subsequent upturns, and so it is important to also examine changing relationships, either graphically or by timevarying parameter models.Â (LWH)
Suggested Reading Cliff, A.D., Haggett, P. and Ord, J.K. 1985 Spatial aspects of influenza epidemics. London: Pion. 
