Predicting GDP with ARIMA Forecasts

Is the U.S. economy headed for a new recession? The risk is clearly elevated these days, in part because the euro crisis rolls on. The sluggish growth rate in the U.S. isn’t helping either. But with ongoing job growth, albeit at a slow rate, it’s not yet clear that we’ve reached a tipping point. Given all the mixed signals, however, forecasting, is unusually tough at the moment. In fact, it’s never easy, but it’s necessary just the same. But how to proceed? The possibilities are endless, but one useful way to begin is with so-called autoregressive integrated moving averages (ARIMA). It sounds rather intimidating, but the basic calculation is straightforward and it’s easily performed in a spreadsheet, which helps explain why ARIMA models are so popular in econometrics. A more compelling reason are a number of studies that report that ARIMA models have a history of making relatively accurate forecasts compared with more sophisticated competition.

As a simple example of the power of ARIMA forecasting, let’s consider what this statistical tool is telling us about the next quarterly change in nominal GDP for the U.S. Making a few reasonable assumptions (discussed below), a basic ARIMA forecasting model predicts that fourth quarter nominal GDP will rise 4.7% at a seasonally adjusted annual rate. For comparison, that’s slightly lower than the government’s initial estimate of 5.0% growth for the third quarter. (Remember, we’re talking here of nominal GDP growth vs. real growth, which strips out inflation. Real GDP is the more popularly quoted series.)

For comparison, let’s compare my ARIMA forecast with GDP predictions via the Survey of Professional Forecasters (SPF). SPF data is available on the Philadelphia Fed’s web site and is reported in nominal terms, thus my focus on nominal GDP. To cut to the chase, ARIMA has a history of dispatching superior forecasts compared with SPF. To be precise, I’m comparing ARIMA forecasts with the mean quarter-ahead prediction of economists surveyed quarterly in the SPF reports.

Ok, let’s take a closer look at the details by reviewing a few basic ARIMA concepts. Keep in mind that in the interest of brevity I’m glossing over the details. For a complete discussion of ARIMA, an introductory econometrics text will suffice. One of many examples: Peter Kennedy’s A Guide to Econometrics. Meanwhile, the main point with ARIMA forecasting is that it’s a tool for using a time series’ history to make a forecast. Yes, it’s naïve, but the fact that ARIMA’s errors tend to be low relative to many if not most other forecasting techniques makes this approach worthwhile. It’s not a crystal ball, of course, and so ARIMA forecasts should be considered in context with other prediction techniques.

The first step is taking the data series (in this case the historical quarterly nominal GDP numbers) and regressing them against a lagged set of the same data. For the analysis here, I’m regressing each quarterly GDP number against 1) the previous quarter; 2) two quarters previous; and 3) four quarters previous. Next, I ran a multi-regression analysis on this set of lagged data to estimate the parameters, which tell us how to weight each lagged variable in the formula that spits out the forecast. To check the accuracy of the parameter estimates, I also ran a maximum likelihood procedure. (As a quick aside, all of this analysis can be easily done in Excel, although more sophisticated software packages are available, such as Matlab and EViews.)

ARIMA’s forecasts are naïve, of course, but based on history it does fairly well compared with SPF. The in-sample forecasting errors (residuals, as statisticians call them) for ARIMA’s average deviation from the actual reported GDP number is a slight 0.0049% since 1970. That’s a mere fraction of SPF’s 3.06% residual over the past 30 years. There are several additional error tests we can run, but the simple evaluations above offer a general sense of how a naïve ARIMA model can provide competitive forecasts vs. the expectations of professional economists.

Alas, like all econometric techniques that look backward as a means of looking ahead there’s the risk that sharp and sudden turning points in the trend will surprise an ARIMA model. That’s clear by looking at recent history, as shown in the chart below. Note that when the actual level of nominal GDP turned down in 2008 as the Great Recession unfolded, ARIMA’s forecasting error rate increased. But ARIMA fared no worse than the mean SPF predictions. In fact, you can argue that ARIMA did slightly better than SPF, as implied by tallying up the errors for each during 2008 and comparing one to another.

Errors are inevitable in forecasting. The goal is to keep them to a minimum, a task that ARIMA does quite well. A certain amount of trial and error is building and adjusting ARIMA models is critical. In fact, the errors can help improve ARIMA forecasts. By modeling the errors and incorporating their history into ARIMA’s regressions, there’s a possibility that we can reduce the error in out-of-sample forecasts going forward. That’s because if you design an ARIMA model correctly, the errors should be randomly distributed around a mean of zero. In other words, errors through time should cancel each other out. In that case, the error terms can be useful for enhancing ARIMA’s forecasting powers, i.e., reduce the magnitude of the errors.

Yes, we still need an understanding of economic theory to adjust, interpret and design ARIMA models for predicting GDP and other economic and financial data series. But considering the simplicity and relative reliability of this econometric technique, ARIMA forecasting is a no-brainer. At the very least, it provides a good benchmark for evaluating other forecasts.