There has been much interest since the 1970s in the question of whether lagged oil price changes help forecast US real GDP growth (Hamilton 2009). This question has taken on new urgency following the large fluctuations in the price of oil in recent years. There is interest not only in the question of possible asymmetries depending on whether the price of oil goes up or down, but also in the idea that increases in the price of oil beyond certain time-varying thresholds may trigger recessions. In a recent study together with Robert Vigfusson, I examine how successful a number of linear and nonlinear models of this type are in reducing the out-of-sample prediction mean-squared error (MSPE) of US real GDP growth (Kilian and Vigfusson 2012).

A useful reference point for this debate is the ability of oil prices to improve on simple univariate autoregressive forecasts of US real GDP growth at horizons up to two years. It can be shown that there are at best small out-of-sample MSPE reductions when forecasting cumulative US real GDP growth from bivariate linear VAR models that include the percent change in the price of oil in addition to real GDP growth. This finding is robust to whether the price of oil is specified in nominal or in real terms and whether the oil price is treated as exogenous or as endogenous with respect to US real GDP. One possible explanation for this result is that the predictive relationship in question is nonlinear. Indeed this possibility has been discussed at length in the existing literature, but the out-of-sample forecasting performance of these nonlinear models has never been evaluated systematically. In fact, suitable econometric models have been developed only very recently.

In this context, Hamilton (2003) made the case that the predictive relationship between oil prices and US real GDP is nonlinear in that (1) oil price increases matter only to the extent that they exceed the maximum oil price in recent years and that (2) oil price decreases do not matter at all. He provided in-sample evidence that including appropriately defined lagged net increases in the price of oil in an autoregression for real GDP growth helps predict US real GDP growth one quarter ahead. This evidence is backed up by our study looking at more recent data (Kilian and Vigfusson 2011). Evidence of in-sample predictability, as documented in these studies, however, need not translate into out-of-sample gains in forecast accuracy, which is the ultimate question of interest to policymakers and applied forecasters.

To resolve this question, it is necessary to evaluate and compare a wide range of out-of-sample forecasting models for US real GDP based on nonlinear transformations of the price of oil that are asymmetric in oil price increases and decreases. A striking result of this comparison is that, among the many alternative asymmetric models that have been suggested in the literature, only a multivariate generalisation of the predictive model proposed by Hamilton (2003) produces systematic MSPE reductions at longer horizons. There is no evidence in support of forecasting models based on the one-year net oil price increase, models based on the uncensored percentage oil price increase, or models based on large percentage increases in the price of oil, in contrast.

The performance of the three-year net increase model in some cases is impressive. For example, based on the three-year net increase in the US refiners’ acquisition cost for crude oil imports, the MSPE reductions are between 19% and 26% at the one-year horizon and between 18% and 17% at the two-year horizon. Similar results are obtained with some other oil price series as well. At the one-quarter horizon, however, the results are less clear cut and depend on the precise definition of the oil price variable.

To date much of the perceived empirical success of the three-year net oil price increase specification has been attributed to the fact that this oil price measure is asymmetric, with little attention to the fact that this definition also embodies other nonlinearities. In this regard, it can be shown that reductions in the MSPE at least as large as for the three-year net oil price increase model can be obtained based on an alternative forecasting model that is symmetric in the three-year net oil price increases and decreases. The results for this *net oil price change* model specification suggest that the asymmetry embodied in the three-year net oil price increase measure is irrelevant for out-of-sample forecasting, if not harmful. This result is consistent with the fact that all other asymmetric specifications considered appear inferior to forecasting models that are symmetric in the price of oil.

The three-year net oil price change model not only tends to be at least as accurate as the corresponding three-year net oil price increase model, but it is more robust to the definition of the oil price variable, more robust across forecast horizons, and more robust to changes in the forecast evaluation period. In short, if there are nonlinearities that matter for forecasting they appear related to how far the current oil price deviates from its most recent extreme values, not to whether the price of oil increased or decreased relative to that threshold. This evidence directly addresses the common concern among many policy makers that the feedback from oil prices to the economy may become stronger once the price of oil passes certain possibly time-varying thresholds. Furthermore, a number of alternative and equally economically plausible symmetric nonlinear specifications (including models that focus on large oil price changes or models that control for time variation in the oil share) cannot replicate the forecasting success of the three-year net oil price change model.

A question of obvious interest is how much of the decline in US real GDP growth during 2008/09 could have been forecast with the help of the three-year net oil price change model. Based on the four-quarter-ahead forecast, further analysis shows that the three-year net oil price change model anticipated about one third of the observed decline in US real GDP in 2008, while linear models essentially failed to predict any decline. These results appear much more plausible than the corresponding forecasts from the three-year net oil price increase model, which imply that virtually all of the 2008 recession could have been forecast one year in advance and that the financial crisis played no role in the 2008 recession. The latter economically implausible result can be traced to over fitting problems in small samples.

In fact, a similar – if much less severe – overfitting problem also afflicts to the three-year net oil price change model. The apparent over fitting may be countered with some simple ad hoc adjustments of the model coefficients. With these corrections, the three-year net change model would have forecast only about 15% of the observed cumulative decline in US real GDP in 2008 one year in advance, which is still much larger than the decline implied by linear VAR forecasts, but more in line with other nonlinear symmetric forecasting models.

These results reinforce a growing body of work that has questioned the role of asymmetries in the relationship between the price of oil and the US economy, while drawing attention to a previously undocumented type of threshold nonlinearity in the predictive relationship between the price of oil and US real GDP. The question of how important these threshold effects are deserves further study on extended samples and on other time series. The preliminary findings in this regard discussed here have potentially important implications for applied forecasters, but also for economists interested in modelling the transmission of oil price shocks. For example, there is no theoretical model to date that would rationalise the type of the threshold effects embodied by three-year net oil price change models.

# References

Hamilton, JD (2003), “What Is an Oil Shock?”, *Journal of Econometrics*, 113:363-398.

Hamilton, JD (2009), “Oil prices and the economic recession of 2007-2008”, VoxEU.org, 16 June.

Kilian, L and RJ Vigfusson (2011), “Are the Responses of the U.S. Economy Asymmetric in Energy Price Increases and Decreases?”, *Quantitative Economics*, 2(4):419-453.

Kilian, L and RJ Vigfusson (2012), “Do Oil Prices Help Forecast U.S. Real GDP? The Role of Nonlinearities and Asymmetries”, CEPR Discussion Paper No. 8980.