The quarterly time series of German consumption and income are analyzed with respect to seasonality and stochastic trends. It emerges that both variables can be appropriately described by a periodically integrated autoregression. An implication is that the stochastic trend and the seasonal fluctuations are not independent for each of the univariate series. In order to test for cointegration across the two series, we propose several methods which take account of the relationship between seasons and trends in the univariate series. Some of these methods boil down to extracting the stochastic trend from the univariate series in a first step and to relating these trends using cointegration techniques in a second step. Another method is an extension of the Johansen cointegration testing approach to periodic vector autoregressions. Monte Carlo simulations are used to evaluate the empirical performance of the various methods. The main empirical result is that only in the first quarter there seems to be cointegration between German consumption and income.

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Keywords consumer economics, seasonality, time series models
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Franses, Ph.H.B.F, & Paap, R. (1995). Seasonality and stochastic trends in German consumption and income, 1960.1- 1987.4. Empirical Economics: a quarterly journal of the Institute for Advanced Studies, Vienna, 20(1), 109–132. doi:10.1007/BF01235160