## Monday, December 26, 2016

### What's the meaning of a 250% change, SA annualised?

by Ajay Shah.

My Business Standard article of today has raised questions about the meaning of SAAR percentage changes.

When we deal with time periods of different lengths, the raw percentage change can be confusing. As an example, if I promise to convert Rs.100 into Rs.101 on a one day horizon, this is 1% return on a one-day horizon. The universal quoting convention is to express this on an annualised basis: $1.01^{365}$ is 3678%. So we would say that this investment opportunity offers 3678% returns.

Similar issues apply when dealing with month-on-month ("m-o-m") changes. If 100 becomes 110 within one month, it's confusing to say that this is a 10% change. It is expressed as $1.1^{12}$, i.e. a 213% change on an annualised basis.

When expressed as continuously compounded returns, the month-on-month annualised change is 1200*diff(log(x)) where $x$ is the levels series. This gives time series of changes which have somewhat better statistical properties for downstream use. For small values the two paths are identical, but for large values the continuously compounded values have fewer extreme values.

How much did exports change in the month of November? Here are the steps through this must be computed:

1. The year-on-year ("y-o-y") change compares November 2016 to November 2015. It is the sum of 12 changes, one for each month. It does not tell us what happened in November 2016.
2. We would like to compare October 2016 to November 2016 to know what happened in November 2016.
3. But there are problems of comparison: Seasonality, Diwali.
4. Seasonal adjustment procedures solve these. They yield a time-series of a seasonally adjusted level of exports.
5. With this, we are permitted to compute the percentage change from October 2016 to November 2016.
6. But this is not comparable with all the percentage changes that we are used to seeing, which are on an annualised basis.
7. Hence, the SA levels series $x$ is converted into an annualised rate of change using the formula 1200*diff(log(x)), which is analogous to $(1+r)^{12}$. This is called the "seasonally adjusted annualised rate" (SAAR) of change.

An interesting property of this way of thinking is that the year-on-year change is the trailing 12 month moving average. It's the average of the changes of the last 12 months. It's a slow and sluggish measure of what's going on in the economy. But the units of the two measures -- the y-o-y change and the m-o-m SAAR -- are the same.

#### One concrete example

We take data for heavy commercial vehicle (HCV) sales from January 1999 to November 2016. The raw values in the latest three months are: September: 28,103; October: 29,515 and November: 21,602. This shows a raw month-on-month decline (non-annualised, non-seasonally-adjusted) of 26.8% in November 2016.

We test for the possibility of Diwali effects in this series and find none. Seasonal adjustment gives SA levels for the latest three months: September: 27,028; October 31,845 and November 27,049. This shows a month-on-month decline (non-annualised) of 15.1% in November 2016.

The SAAR is 1200*diff(log(x)) where x is the seasonally adjusted levels. This yields the values of +196.8 in October 2016 and -195.9 in November 2016. Specifically, the latter value, -195.9, is 1200*(log(27049)-log(31845)).

#### Conclusion

When dealing with events like October 2008 and November 2016, this is the essential statistical technology for thinking about what is going on. We don't know of anyone working with Indian data, outside of NIPFP, who has this machinery working. Some people are tossing data into black boxes (e.g. Eviews) for seasonal adjustment, which is better than doing nothing, but not as good as the full thing.

LaTeX mathematics works. This means that if you want to say $10 you have to say \$10.