One of the advantages of blogging: I can pontificate about factual matters without any actual evidence. Here, I will pontificate about whether the paradigmatic example of a Giffen good -- potatoes in 1840s Ireland -- is not actually a Giffen good.
The subject of Giffen goods came up recently when Tyler Cowen said prostitution might be a Giffen good, since prostitutes found they got fewer customers when they offered discounts. But as David Masten noted, the mere fact that quantity demanded falls when price rises doesn't make a good Giffen. Why not? Because the characteristics of the good in question must remain constant, so that the price change is the only causal factor. But in the case of prostitutes, the lowering of price signals a change in what's being offered. David puts it nicely: "When the lady drops her rates from $250 to $20, the customer is wondering if she has contracted an STD, [or] she is falling in love with him and complicating the arrangement… he is no longer sure of which good he is getting."
A similar objection applies to other suggested Giffen goods, such as dog food. Because people want the very best for their pets but (shockingly) are unwilling to try out the pet food themselves, they may take price as a signal of quality. When Purina raises its price, people think Purina is better quality dog food and buy more.
So a Giffen good is not simply a good with a positive relationship between price and quantity demanded; it's one with that relationship for particular reasons. First, the good must be an inferior good (one that you buy less of when your income increases, and vice versa). Second, it must constitute a large enough portion of your expenditures that a change in its price would have a substantial effect on your real income. That is, a rise in its price would make you markedly poorer, even if your nominal income were unchanged. If these characteristics both hold, then a rise in price could -- by making you poorer -- cause you to increase your consumption of the good.
The original and most famous example of a Giffen good is potatoes in 1840s Ireland, during the Irish Potato Famine. A pestilence drastically reduced potato production, thereby raising the price of potatoes. Robert Giffen allegedly discovered that Irish consumption of potatoes subsequently rose, thus violating the law of demand. Why? The usual story is that (a) potatoes were inferior (people will switch from potatoes to meat, fruit, and vegetables as their income increases) and (b) potatoes constituted a large portion of Irish households' expenditures. When the price of potatoes went up, their real income fell, so they cut back on the luxury food items and started buying more potatoes.
Now, here's the part where I pontificate without evidence. I suspect that this most famous of Giffen goods was not, in fact, a Giffen good. Why? Because the Irish, in addition to being potato eaters, were also potato farmers. The potato blight wiped out a large portion of their livelihood. The remaining potatoes sold for a higher price, of course, but I imagine the overall effect on farm revenue was still devastating. And if I'm correct in this supposition, that means that Irish people's real income decreased for a reason other than the rise in the price of potatoes. Even if the price of potatoes had remained constant, the Irish would have bought more potatoes simply because it's an inferior good.
What would be the test to see if my hypothesis is correct? Go back to Giffen's original data and see if you can distinguish between farmers and non-farmers. The non-farmers' income would not have been affected to the same degree (though lower income from business with the farmers might have had an effect). If their consumption of potatoes decreased or remained constant, we could conclude that potatoes were not, in fact, a Giffen good.