In applied econometrics researchers often infer the relation among nonstationary time series by regression of their differences. The aim of this paper is to show that in some circumstances regression of differenced time series tends to reject the relation among their levels. This phenomenon is known as type I spurious regression. Time series are dynamic processes, and the ignored system dynamics will become the systematic errors in regression equations. Differencing does not preserve the underlying relation among time series in regression due to systematic errors. This paper will outline how regression of differenced time series tends to reject the relation between their levels, and so potentially to incur type I spurious regression.

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This paper seeks to model the adjustment process in the stock market by a continuous time state space model focusing on input-out relations. The value of the S&P 500 is generated as the output of the model with earnings and the interest rate as input. The model is found to fit the data well, and indicates that the stock price dynamics can be considered as a price-following-value process. The value determines the time varying trend of price, and random buy-sell pressure drives price fluctuations about value. The 1987 stock price bubble shows up clearly as a gap between price and value.

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Accounting and finance professionals have empirically known that in the long run stock prices are roughly proportional to earnings. However, econometric testing could not been able to verify this expected contribution of earnings to stock prices, thus formed the price-earnings (PE) puzzle in the accounting literature. This paper seeks to solve this puzzle by allowing the earnings response coefficient to be a variable instead of a constant, and shows that the PE puzzle turns out to be a phenomenon of type I spurious regression in econometrics.