20.5.2 Cointegration between Forecasts and Outcomes

Under MSE loss we expect that the forecast errors $y_{t+h}-f_{t+h|t}$ are martingale difference sequences with respect to information observed at time t, i.e., $E[y_{t+h}-$ $f_{t+h|t}|Z_t]=0$ . When ${\gamma }_t$ has a unit root, this implies that $f_{t+h|t}$ must also have a unit root and the series should be cointegrated with cointegrating vector $(1,-1)$ . Some authors have used this implication to test for cointegration between outcomes and forecasts. Liu and Maddala (1992) and Aggarwal, Mohanty, and Song (1995) examine this in applications to exchange rates and macroeconomic data, respectively. Cheung and Chinn (1999) examine cointegration between macroeconomic outcomes and forecasts without imposing the cointegrating vector.

Such tests are unlikely to shed much light on whether the forecasts are efficient. For the forecasts and outcomes not to be cointegrated with the expected cointegration vector (1, −1), the forecasts and outcomes would have to diverge over time. For nearly all forecasting problems we either observe ${\gamma }_t$ or a close proxy of it, so it is difficult to imagine that this could happen in practice. In this sense, the cointegration test does little to separate good from bad forecasts. For example, simply setting $f_{t+h|t}=y_t$ results in a forecast that satisfies the cointegrating restriction and the null of cointegration allows for a lot of serial correlation in the forecast errors, so the test would fail to reject for models that fail the orthogonality conditions described in chapter 16.

20.6 CONCLUSION

Forecasts based on estimated parameters of a model fitted to highly persistent data do not necessarily dominate forecasts of persistent data that impose a unit root. One reason is that the conventional autoregressive model fitted to persistent data is not necessarily the right model for departures from a unit root. Persistence could also be due to fractional integration, stochastic breaks, time-varying parameters, or some other form of model instability.

Considerations such as these have lead Stock and Watson (2003b, pages 466–467) to conclude, “The most reliable way to handle a trend in a series is to transform the series so that it does not have a trend. . . . Even though failure to reject the null hypothesis of a unit root does not mean the series has a unit root, it still can be reasonable to approximate the true autoregressive root as equaling one and therefore to use differences of the series rather than its levels.”

Forecasting Nonstandard Data

ous exception being the binary case covered in chapter 12 which is sufficiently simple and tractable to warrant special treatment. There are other interesting cases of special interest. One example is count data for which the dependent variable is restricted to take on positive, integer values. This case requires its own class of prediction models and distributional assumptions—typically some variation on a Poisson or binomial model—and is covered in section 21.1. Another example is duration data which involves predicting the time before some event happens—e.g., the end of a recession or a change in the Federal funds rate. Here the outcome can follow a continuous distribution, but one that is restricted to only positive values. We cover such models in section 21.2.

Another complication arises due to data revisions. So far we have taken the information set $Z_t$ used to generate forecasts as given and largely ignored complications arising from which pieces of information forecasters have access to and which pieces they do not observe. Section 21.3 considers practical issues related to the construction of forecasts with data that are in some ways irregular and thus require careful attention before being used by forecasters.

Data revisions is a key issue that has generated considerable interest among macro forecasters. Key variables such as GDP growth, industrial production, unemployment, and consumption expenditures are all subject to revisions. Data repositories are now being created for a number of variables to keep track of such revisions. Data revisions can affect forecasts through the conditioning information set, parameter estimation, and even model selection.

Economic data are published at different frequencies: GDP growth is published quarterly, industrial production is available on a monthly basis, payroll numbers are available weekly, while financial data such as stock prices and interest rates are available on a second-by-second basis. A set of recent papers have addressed how to construct forecasts with data observed on different dates and at different frequencies, e.g., at daily, weekly, monthly, or quarterly intervals. This gives rise to so-called mixed data sampling techniques as well as applications of the Kalman filter which can handle such issues. It also raises the issue of how we produce the best estimate— or summary measure—of the current state given the wealth of information observed at one point in time. This is an area commonly referred to as nowcasting and opens up the possibility of constructing a daily measure of the business cycle. We cover this topic in section 21.4. Section 21.5 concludes.