## Friday, April 15, 2011

### Price Indices From Regression Models

We're all familiar with index numbers. We encounter them every day - perhaps in the form of the CPI or the PPI;  or maybe in the form of some index of production. The effective exchange rate is an index that I'll be posting about in the near future. Share price indices are also familiar, although the DJIA has some very peculiar aspects to its construction that deserve separate consideration on another occasion.

The thing about an index number is that it has only ordinal content. That's to say, if  a particular price index, say P, has a (unit-less) value of 110, that number tells us nothing about the price level at all. It's only when we compare two values of the same index - say, the values in 2010 and in 2011 - that the numbers really mean anything. If P = 100 in 2010 and P = 105 in 2011, then the average price of the bundle of goods being measured by P has changed (risen in this case) by 5% - not by \$5 or some other value. In other words, over time, or perhaps across regions, an index number measures proportional changes.

When any index number is constructed, a base period and a base value must first be chosen. For example, we might decide to choose a base year of 1996, and a base value of 100. There's absolutely nothing wrong with choosing a base value of, say, 167.5290 in 1996 - it would just be unnecessarily inconvenient. In that case if the index rose to 184.2819 in 1997, this would imply a relative price change of 100*[(184.2819 - 167.5290) / 167.5290] = 10%. Wouldn't it have been easier if we had chosen the base value to be 100, observed a value of 110 in 1997, and then been able to see immediately that this implied a 10% increase in prices over this one-year period?

Of course, it's the fact that an index measures only relative changes over time that enables us to "re-base" (change the base year) an index without losing any information at all. The numbers in the index series just get scaled, multiplicatively, by the same factor, leaving relative values - and the implications for price changes -  unaltered.