To be submitted by March 15.

Home assignment 11. Non-stationary Time Series. Unit root tests. Cointegration.

To be submitted by March 15.

1. Perform 3 Dickey-Fuller tests and augmented Dickey–Fuller tests for stationarity of the logarithms of expenditure on your commodity (not FOOD or HOUS, file demand_59-03.wf1), disposable personal income and relative price series. Interpret the results, comment also on correlograms. Investigate these series for difference stationarity and trend stationarity. Are these series random walks (with or without drifts), do they include time trends? How did you choose the number of lags in AR process?

2. A researcher has time series data for aggregate consumption, C, and aggregate disposable personal income, Y, for a certain country. She establishes that the logarithms of both series are I(1) (integrated of order one) and she correctly hypothesizes that the long-run relationship between them may be represented as

(1)

where l is a constant and v is a multiplicative disturbance term. It may be assumed that log v is normally distributed with zero mean and constant variance.

The researcher believes that log C and log Y are cointegrated. How should she demonstrate this?

The relationship implies that the long-run growth rate of consumption is equal to that of income. Explain whether it is correct to describe the growth rates as being cointegrated.

The researcher is also interested in the short-run dynamics of the relationship and correctly hypothesizes that they may be represented by the relationship

(2)

where e_t is identically and independently distributed and drawn from a normal distribution with zero mean. State the restriction that has to be satisfied by the parameters if the short-run relationship (2) is to be compatible with the long-run relationship (1).

Show how the restricted version of (2) may be reparameterized as an error-correction model. Explain why fitting the error-correction model, rather than (2) directly, avoids a potentially important problem.

3. Run a double logarithmic regression of expenditure on your commodity on disposable personal income and relative price, plot the residuals, comment. Test the residuals for cointegration (use the table from the lecture slides). Fit an error-correction model for your commodity, assuming that a cointegrating relationship has been found. Interpret the model and its coefficients. Explain the short-run and long-run characteristics of the model.

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