The advance (first draft) of real GDP growth for 2014 I came out a few weeks back. It was extremely low: an annualized rate of 0.1%.

The common explanation for that was the rough winter in the eastern half of the country (where about 3/4 of the real GDP is generated. Fair enough.

There were also assertions that the economy would bounce back, as production that was delayed in quarter I was made up for in quarter II.

That’s more or less what MA processes in estimates are meant to catch. AR processes capture the persistence and slow decay of shocks (they can also cover the much rarer case of swings back and forth). MA processes capture shocks that quickly dissipate or which are reversed.

So, whether they know it or not, pundits who argue that real GDP will rebound in 2014 II are implicitly claiming that real GDP has an MA(1) component, and that it’s pretty strong.

We like ARIMA(p,d,q) modeling because it’s flexible and it works well. But, there are typically a large handful of ARIMA processes that will fit a data series relatively well, with combinations of p, d, and q in the 0, 1, or 2 range.

In class, we’re limited by Excel, so we can only do ARI(p,d) or ARIMA(p,d,0) processes. Within that constraint, and ARI(1,1,) or ARIMA(1,1,0) fits real GDP pretty well. But, with better software, most macroeconomists would argue that an ARIMA(1,1,1) fits the log of real GDP well. That part, at least, is consistent with pundits’ positions. But to answer whether that MA(1) component in strong, and works to reverse shocks, requires estimates.

With quarterly data from 1947 I to 2014 I, an ARI(1,1) estimate looks like this:

ΔY_{t} = 0.008 + 0.374Δy_{t-1} + residual_{t}

Here, y is the natural log of real GDP. This is analogous to the model we estimated in class (and in the handbook) with annual data. The fitted value from this model will have two components from the RHS, while the residual is the difference between the actual and observed values. Like so:

Quarter | ΔY = | Mean + | from AR(1) + | Residual |

2013 III | .0101 | .0050 | .0023 | .0029 |

2013 IV | .0065 | .0050 | .0038 | -.0023 |

2014 I | .0003 | .0050 | .0024 | -.0071 |

Do note that this is for quarterly logged data. Because the data is quarterly, we need to multiply it by 4 to annualize it. Because it is logged and in decimal form, we need to move the decimal point two places to the right to convert to percentages.

So, for the second row, we got annualized growth of 2.60% in 2013 IV. Of that, we get 2.00% from the mean, 1.52% from past positive shocks that were embodied in past real GDP and whose persistent effect is captured by the AR(1) term. But, we subtracted off 0.92% because of a negative shock in that quarter.

But things were worse in 2014 I. The mean was the same (the mean always is), but the contribution from last period’s low growth was a weaker 0.96%. Since we expect residuals to equal zero (because this is before we know what they are), we would have forecast growth of 2.96% for 2014 I. But, when it came in much lower, it was from a residual that we can’t explain (other than to tell a story about bad weather).

How bad is that residual? Well, bad, but not horrible: it’s percentile would be in the high teens. We hadn’t gotten a negative shock that bad in 6 quarters, and (befitting the weakness of our economy over the last several years) we haven’t had a positive shock of that magnitude in about 10 years.

The thing is, residuals are unlikely to be repeated (recall that if they have a pattern, we’ve done something wrong). So our expectation now of the residual for 2014 II is still zero.What does the ARI(1,1) forecast for growth in 2014 II. Here goes:

Quarter | ΔY = | Mean + | from AR(1) + | Residual |

2013 III | .0101 | .0050 | .0023 | .0029 |

2013 IV | .0065 | .0050 | .0038 | -.0023 |

2014 I | .0003 | .0050 | .0024 | -.0071 |

2014 II | .0051 | .0050 | .0001 | .0000 |

That forecast for 2014 II works out to just a tad over 2.0%. Anything better than that indicates a positive shock. Anything worse indicates another negative shock. Why is growth forecast to be so weak? Because the AR process looks back at lagged output rather than the lagged residual. There’s no sense in which the AR process gets a bad shock and says “Oh … we’ll claw some of that back the next period.” Instead, the AR process looks back and says that the growth rate was low, and that’s going to persist until we get a positive shock to break us out of our funk.

Now let me show you what an ARIMA(1,1,1) estimate looks like, and how much claw back it says we’re likely to expect. The estimate is:

ΔY_{t} = 0.008 + 0.501Δy_{t-1} + residual_{t} - 0.146*residual_{t-1}

This model is a little richer: there’s a little more persistence to approximate growth rates through the larger AR(1) coefficient, but there’s also a tendency for some of the effects of shocks to be reversed. To see this, not the negative coefficient on the last term. This means that a positive shock now will be reversed by 14.6% the next quarter, and vice versa for a negative shock.

Here's how the three fits and forecast work out:

Quarter | ΔY = | Mean + | from AR(1) + | Residual + | MA(1) |

2013 III | .0101 | .0040 | .0031 | .0032 | -.0001 |

2013 IV | .0065 | .0040 | .0051 | -.0021 | -.0005 |

2014 I | .0003 | .0040 | .0032 | -.0072 | .0003 |

2014 I | .0051 | .0040 | .0001 | -.0000 | .0011 |

The mean is smaller for this model. That doesn’t mean it’s absent, just that it’s being picked up by other factors. The AR contribution is a bit larger from this model because the AR coefficient is larger. The residuals are very close to what we got with the ARI(1,1) model: this is the sense in which the improvements between one ARIMA model and another are not that large.

The last column is the new one, and it shows the amount of claw back of negative residuals. And, for that last row, it indicates that of the large negative shock in 2014 I, that drew down the growth rate by 2.88%, we will get back about 0.44% in this quarter.

In fact, most of the sense in which this quarter is likely to be weak is the AR component stretching out the weak growth rates of the last couple of quarters. Persistence alone will draw down the growth rate by 1.24% in this quarter: about 3 times the size of the claw back from the MA(1) term.

So, the MA terms are important in modeling real GDP, but not big enough to be the primary part of the story.

And, with either an ARI(1,1) or an ARIMA(1,1,1) we get to roughly the same conclusion: the large negative shock in 2014 I will cause 2014 II to be weak as well, with a forecast of 2.0% growth.

Do keep in mind that one of the messages of this course is that we need to be comfortable with more volatility in macroeconomic outcomes than politicians and pundits are going to tell us. So given that the 95% confidence interval on real GDP growth rates is roughly ± 2%, our forecast for 2014 II is something like 0-4% rather than the more typical 1-5%.

It's September 26th, and the revisions for 2014 II are all done now.

ReplyDeleteThe advance estimate came out with growth at 4.0%, the revision was at 4.2%, and the final estimate was 4.6%. Those are big numbers, on the outside of my 95% confidence interval.

What does this mean?

Well, it's a lot larger than the clawback that we'd expect of the growth lost in 2014 I. Roughly, the model predicted growth of about 1.6%, plus 0.4% clawed back, and no autoregressive contribution at all because the economy's been weak for a while.

More importantly, it means that we had a big positive shock in 2014 II: on the order of 2.6% (or 0.0065 in the last table).

There's considerable optimism in the legacy media that this foretells strong growth for the rest of 2014, but it doesn't mean that at all. By definition, a big positive shock is unlikely to be repeated. It's a shock, after all.

So, what can we expect from the ARMA(1,1) in growth rates? Going from the left, the actual entry form 2014 II (note the typo in the row label) will be .0115 instead of the forecast of .0051. In 2014 III, we'll then be forecast to get the 0.0040 mean, plus a little over half of the 2014 II growth for .0058, with the MA coefficient knocking off about .0009. Add them up, and the model forecasts 2014 III at 3.6% after annualization.