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When we were trying to decide whether to buy or rent in late 2010 I read a bunch of the literature on housing prices (and ran VARs on Victoria prices). My take is that the literature is pretty consistent that they're more forecastable than financial assets due largely to transactions costs and their dual role as consumption goods, but that a martingale is still a pretty good approximation. There is reasonably strong evidence, though, for long term negative serial correlation in returns and short term positive correlation (.e.g, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2199115).

That said, I would like to see some good papers trying to explain the market in Vancouver.

I think the math problem is actually rather hard, but more needs to be specified about the assumptions before it can be solved. Do you mean housing prices follow a non-drifting random walk with iid normal innovations?

Chris: "Do you mean housing prices follow a non-drifting random walk with iid normal innovations? "

Yes, I think that's what I mean. Definitely non-drifting (for real house prices, because nominal should drift at 2% given the Bank of Canada's target). Definitely iid innovations. Normal if necessary, but not necessarily normal.

Because I am trying to model what total ignorance (except for the Bank of Canada's inflation target) would say, so I can use it as a benchmark to test whether people who claim to know something actually do know something.

My math/stats is bad. I was trying to figure out the case where each period there is a 50% probability of a +1 shock and a 50% probability of a -1 shock. And calculating the probability the price would be less than or equal to the original price before T periods had passed. It looked to me like it was converging towards 100% as T got bigger.

Just on that last comment, wouldn't nominal asset prices be expected to drift along with nominal incomes, not the CPI?


I'm worried about your assumption of no-drift in house prices. Obviously the drift rate matters for the probability that you eventually see a drop.

I'm pretty used to the assumption of positive economic growth and (in Canada) positive population growth as well. I would have thought that rising population increases (ceteris paribus) the demand for housing and increases the economic rents associated with centrally-located land. I would also have thought that rising incomes would additionally increase the demand for housing.

Why do you think the drift rate should be zero?

BTW, I think the problem you're trying to solve is close to that of determining the price of an American put option on the house with a strike price equal to the current market value. An primer on option pricing should give you some traction on the math if you want it.

Okay, now I've read your second random thought. The math is ugly because it is an American option, but your intuition is correct for the case that you examine. (More generally, for a driftless martingale, the probability that you eventually cross ANY threshold converges to 100%.)

If you really want to do the option pricing, you really need to worry about both the variance of the shocks and the drift rate of house prices. Good luck with that.

As for the third random thought, where does the assumption that Canadian housing prices should (eventually?) move with US housing prices come from?

Is that true historically? approximately?
Would you think that is a reasonable way to think of the relative cost of housing in Vancouver and Montreal?

most who quote house price changes do not consider deterioration of the structure and its equipment...i have put two furnaces and two stoves in 41 years in this house, and just last week i paid more than two thirds of what i originally bought this house for to have the roof replaced...

Britmouse: If people spend a fixed percentage of their income on houses, we would expect to see the average house price rise with nominal income. But some of that price rise would reflect a rise in the quality or size of house, because people will but bigger and better houses as they get richer. Teranet-National Bank data (or Case-Shiller) tries to adjust for changing quality/size by looking at repeat sales, so it's measuring pure house price inflation.

Simon: "Why do you think the drift rate should be zero?"

I'm not sure I do think that. Personally I would expect a slight positive drift, because of rising population and fixed land, and maybe technical change in house production being slightly slower than average? But what I am trying to do here is establish some sort of null hypothesis to reflect the views of someone who confessed total ignorance of the determinants of house prices, and had zero ability to forecast them. So that we have some sort of absolute benchmark against which to test the claims of pundits who claim to know something. "Can you predict house prices better than a complete ignoramus?" First we have to say what a complete ignoramus would predict.

"... but your intuition is correct for the case that you examine."

I am relieved to hear that! I wasn't sure.

The American option is a great analogy. I hadn't seen that.

"As for the third random thought, where does the assumption that Canadian housing prices should (eventually?) move with US housing prices come from?"

Nowhere really. Just the idea that Canada ought to be like the US, except when we have reasons for believing it won't be. But you have presumably noticed people saying that the US had a house price crash, therefore Canada will have one too? In other words, I think that the idea that the US will converge to Canada about as plausible as the idea that Canada will converge to the US.

rjs: I was (implicitly) excluding maintenance and depreciation from house prices. Thinking in quality-adjusted and inflation-adjusted terms.

Nick, if I understand correctly (and apologies if I'm not), you're basically asking about the expected first passage time of a no-drift random walk? Or some other related property, such as the probability that the walk will be lower T time units in the future than it is today?

I'd think that this has been worked out already somewhere. Probably worked out independently in various fields where random walks are of interest. Evolutionary biologists and population ecologists both care about random walks, because they're interested in whether a rare allele, or a rare species, will drift to extinction within T years, or drop below some specified frequency or abundance at some point in the next T years.

Have you already tried googling "random walk properties" or "random walk first passage time" or something?

Jeremy: Aha! I Googled. I think what I am talking about is a particular application of "gambler's ruin". A gambler playing a fair game will eventually lose his whole stake with certainty, if he plays long enough. Some theorem by Christiaan Huygens.

"Or some other related property, such as the probability that the walk will be lower T time units in the future than it is today?"

Not quite. We know that there is a 50% probability that a random walk will be less than today T periods from today. What I need is the probability that it will be less than today at *some time between* today and T periods from today. In other words, if I said "house prices will be lower than today some time in the next 10 years", what is the probability I will be right, if house prices follow a random walk?

"An alternative narrative is that US house prices were not too high in 2006, but were too low in the recession, and that Canadian house prices are not too high. Canada came back up to reality sooner, but the US will eventually follow."

Friedman's plucking model?

In the countryside prices should rise with the cost of construction, labor and commodities, roughly cpi, while large urban areas where construction opportunities are limited should rise with incomes, population and growth, but if those begin to lag, prices will as well. More productive labor and building technologies help keep construction costs down. Improved transportation and communications can decrease the pressure on urban areas allowing them to expand while increased congestion and transport costs can increase the pressure.

"It would only have been a mistake if you could have rented a house for less than the foregone interest and other costs of buying the house."

Risk weighted cost. There are always risks involved with tying up a major portion of your net worth in a single potentially illiquid asset, and there are opportunity costs.

You have to consider what the chances are for you house to go under water and for how long. Owning an underwater home can mean not only a loss of equity, but also being unable to sell without paying off the negative equity, and, until recently being unable to refinance. It also does bad things to your credit rating and means you would have to reject any job offers that would require moving.

Was there a nationwide housing bubble? It's hard to make a conclusive case for one.

Were there serious local bubbles in certain markets to the extent that they had economic significance? Probably yes.

This is Las Vegas

Price to rental ratios of over 70 in sections of Vancouver strain the concept of rationality.

But "markets can remain irrational longer than you can remain solvent." Part of the reason is that short positions are more difficult and dangerous than long positions in many markets. This is probably intentional, since it is well known that shorts are the spawn of the devil.

Also behavioral economics would seem to indicate that people are more easily motivated by fear and greed than rational argument and sound calculation. As a result bubbles often suddenly pop. Since fear is stronger than greed, and the financially overextended have much to fear, the crash will be rapid and likely overshoot, as well.

An image of such a "Minsky moment":

"Clearly we need a date, but we also need a probability. And the later the date, the higher the probability would need to be."

Your neglecting predictions like:

I predict that sometime between 1JUN15 and 1JUN18 Vancouver price to rental ratios will strongly revert to the mean to less than 60% of their current values averaging over all areas where the current ratio is 30 or greater and that most of the adjustment will be to price and not rent.

I'm not predicting this, though I expect it's more likely than not to be correct, which isn't good enough.

Also consider that a bubble can be a diagnosis as well as a prediction. After a bubble has popped, we have much more evidence. We can see that the fall in prices was rapid, We can see that the last leg of the increase was Ponzi financed. We can see whether there was rampant fraud and misrepresentation.

A true bubble is irrational, and the above are evidence of irrationality.

OTOH the internet bubble is a hard call. It's hard to make a case that the long term consequences were bad. A great deal of technology was created that turned out to be valuable. Someone who bought Amazon rather than pets.com did fine.

Perhaps there's a distinction to be made between irrational or fraudulent enthusiasm and enthusiasm that is merely excessive and somewhat premature.

The British railroad bubble of the 1840s presents similar questions.

I was trying to "use math" starting with the most simple case - imagine that when price drops it always drops to zero never to recover again. Then the probability really looks like your graph, which is actually the function f(x)= 1-0,5^x

Then I tried some different parameters and found out that the behavior of the function depends on volatility (how much the price drops for a given period). When I wrote it all on paper it started to reminded me the binomial tree that I used for some stuff back at university. So I googled it and it seems that there actually exists a thing called "binomial pricing model". I think this is the best place to start looking for answers.

JP: "Friedman's plucking model?"

Aha! Yes. What goes down must come up, rather than what goes up must come down. It's funny we don't have a simple clear word for a negative bubble.

Lord: "More productive labor and building technologies help keep construction costs down."

Yes, but my hunch is that technical change and productivity growth in building houses has been slower than for goods in general. And if so we should see houses prices rising relative to goods in general. To my untrained eye it looks like they build houses today in pretty much the same way they built them decades ago, while other sectors of manufacturing have changed much more. Can anyone who has a trained eye confirm or deny? People live in century-old houses; nobody uses decade-old computers.

Peter N: "Risk weighted cost. There are always risks involved with tying up a major portion of your net worth in a single potentially illiquid asset, and there are opportunity costs."

Here's an old post of mine where I take an opposite view. The idea is that we are born with a short position in housing (because we need somewhere to live) and that by buying a house we are reducing risk by covering that short position.

"Your neglecting predictions like:

I predict that sometime between 1JUN15 and 1JUN18..."

Yep. If someone predicts house prices will be lower at date T+X than at date T then that would have a 50% probability of being true in a random walk model. So anybody who makes a prediction like that only needs to assert it with greater than 50% probability to be making a substantive prediction. But most forecasters don't seem to make predictions like that, I think.

JV: I googled binomial pricing model, and I think you are right. It looks closely related.

House construction began to really change maybe 15 or 20 years ago, and this is accelerating. Houses may do mostly the same things for people, but they won't be built the same way. For instance the fluted columns on my porch are made of urethane, cost $90 apiece and will last forever. and they need to be repainted less often.

The originals were made by a carpenter's assistant with a molding plane. It must have taken a while, but he was only being paid $1.50 a day or so.

It's still early days for this, as it is for medicine. Come back in 20 years, and you won't recognize the place (metaphorically, anyway).

"The idea is that we are born with a short position in housing (because we need somewhere to live) and that by buying a house we are reducing risk by covering that short position."

Fine, but this is the riskiest and most inflexible way of doing this there is. You could rent and get the same cover with securities.

Peter N: "You could rent and get the same cover with securities."

But those securities would need to be indexed to rents in the area where you live. You would have to buy shares in local rental housing companies. Or you could buy a house, and avoid all the moral hazard problems by being your own tenant, and being your own landlord.

Yes, the illiquidity is a problem, if you might want to move house.

"It looked to me like it was converging towards 100% as T got bigger."

I'm not clear on what math there really is to do. Anyway, you've got it right. Since we're talking about convergence in the number of time steps of your binomial model, we might as well deal with the continuous-time limit which is just GBM with mu zero and sigma some unspecified positive value.

We are interested in the time-expectation of a single realization of this path, which is exp[(mu - sigma^2/2)t]; in our case that simplifies the return to -t.sigma^2/2, which diverges to -infinity, implying that the expectation of house prices converges to zero.

Simon objected that mu might not be zero; well, when mu > sigma^2/2, the expectation will diverge but a single path will still fall arbitrarily low with probability one in infinite time.

Finally, note that if you could take the ensemble average (over infinitely many realizations) instead of the time average, the realized return would be mu with probability 1. That is because the ensemble expectation is dominated by a vanishingly small number of diverging realizations - the same as the St. Petersburg lottery, for example.

If we know nothing about house prices, we are in a situation like the following. A Martian walks up to you and tells you a Martian house costs 100,000 galactic credits. Is this too high or too low? Since you have no idea what the wages on Mars are, or the prices of other goods and services, saying they could go up or down is about all you can do. Therefore, I think I agree with your arguments in this post, as you stated them.

But most people who complain about house prices have usually used some information about relative prices when making their arguments (including the house prices to rents or wages). Housing is a significant portion of the economy, so there are limits on what can happen.The most important constraint being how much new buyers can finance out of their income. If this hits its effective limit, the price process should have a bias towards falling prices. (I am not saying that limit has been hit in Canada, but we are at least approaching it.)

I think housing prices will tend to follow wages (or even rise faster), which are normally above CPI inflation. As people get richer, they have spent a higher proportion of their wages on housing. As you note above, some of that reflects a bigger and better house, and so some of the increased spending should drop out of a quality-adjusted housing index. But some of that increased spending will be the land cost, which will increase the value of a quality-adjusted house price index as well. As such, house price indices could have a drift that is greater than CPI (well, that was the experience of recent decades).

In any event, I think the more interesting question from a macro point of view is: will Canadians continue to build new housing units at the same pace? If construction drops back to a more "typical" level, that will create a big hole in construction employment. Where do those people find jobs in the short-term, considering that the country will be losing a debt-fuelled source of final demand at the same time?

Phil: I confess I understood little of your comment (my fault) except this bit "Anyway, you've got it right.", which I was very pleased to read. Does "GBM" stand for "Gaussian Brownian Motion"?

Brian: I really like your Martian thought-experiment. That captures exactly what I had in mind.


Sorry for the jargon. GBM stands for geometric brownian motion, the continuous time limit of your discrete time model of fixed geometric returns.

What I was saying was much simpler than it must have appeared. It is just that when taking expectations, one can do so with respect to "states of the world" (ensemble) or over time in a single, arbitrary state of the world. Sometimes it is appropriate to do one and sometimes the other and sometimes it doesn't matter. For example of the latter, suppose we flip a coin with payoff a for heads and b for tails. It doesn't matter whether we flip a bunch of coins at once or one at a time over an interval, because the cumulative payoff is arithmetic, not depending on time. But if the payoff is 2X for heads and X/2 for tails with X the running stake, then it makes a big difference.

In your model, the ensemble probability that house prices will drop (or rise) is zero. But it would be wrong for our perfectly ignorant but perfectly rational prospective home-buyer to use ensemble expectations because that is not the problem she faces. She will live in only one world and she wants to know how likely it is that prices will fall below their current value in her world. The answer is 100%, because if we choose a state of the world at random then we will choose one in which prices drop at some point with probability one.

This might seem paradoxical at first, but the resolution is that fraction of states of the world in which house prices do not drop grows arbitrarily close zero, but at the same time house prices in those states are growing arbitrarily high - the two effects balance in the ensemble expectation. It's the same reason why a hyper-rational risk-neutral player would only pay a finite (and rather small) sum for a ticket in the St. Petersburg lottery: although the ensemble expectation of the payoff is infinite, the expected payout of one game (or any finite number of games) is finite.

> In your model, the ensemble probability that house prices will drop (or rise) is zero.

And, I should add, the ensemble expectation of the waiting time until house prices drop (or rise, but not "drop or rise"!) is infinite.

"That means they are equally likely to rise (say) 10% next year as they are to fall 10% next year. And equally likely to rise 20% next year as fall 20% next year. Etc. For any year, regardless of anything. I think that is what someone who admitted total ignorance would assume."

What you are assuming is that prices tend towards zero. E. g., suppose that prices start at 100. Next year they are 10% higher, or 110. The following year they are 10% lower, or 99, a loss of 1%. Well, maybe order matters. Suppose that the second year they are 10% lower, or 90, and the third year they are 10% higher, or 99.

In fact, the opposite should be the case. Prices should tend higher. Why? Because there is a lower bound but no upper bound. That's why evolution appears to progress towards increasing complexity.

"In fact, the opposite should be the case. Prices should tend higher. Why? Because there is a lower bound but no upper bound. That's why evolution appears to progress towards increasing complexity."

Min, that was beautiful.

Phil: thanks. I think I'm beginning to get my head around it now.

Sina sent me a good email, which I'm also starting to get my head around. Sina wants to change the question from: "what is the probability, as a function of T, that house prices will be lower than present before T periods have passed?" to "what is the probability, as a function of T and K, that house prices will be K% lower than present before T periods have passed?" My red curve now becomes an S-shaped curve, that starts near 0% for T=0, then rises quickly at first, then inflects and rises more slowly, and asymptotes to 100%. (And I think my red curve shouldn't start at 50%, unless we have discrete time, and the initial period in my graph is T=1.)

Min: I was worried about the lower bound of $0. That's why I specified *log* price. Does that get around it?

(That evolution thing was beautiful. I wonder if Jeremy Fox is reading?)

"what is the probability, as a function of T and K, that house prices will be K% lower than present before T periods have passed?"

If you permit me to use the continuous-time version again (GBM, so log prices follow a Brownian motion) then the answer is that the first hitting time of the price L=(1-K)S, with S the current price, is distributed according to the inverse gaussian distribution with parameters:

IG((ln(L)-ln(S))/(mu - sigma^2 /2), (ln(L)-ln(S))^2/sigma^2)

where mu is the annualized drift (zero here) and sigma the annualized volatility. If you insist on the discrete time version you have to count, which too fussy for me to try here.

Nick Rowe: "I was worried about the lower bound of $0. That's why I specified *log* price. Does that get around it?"

I think that's what you want. Phil Koop looks like the authority here. :)

Nick Rowe: "That evolution thing was beautiful."

Glad you like. :) It's not original, though. I think that Gould pointed that out. I doubt if it was original with him, either.

Matt Yglesias:
I find this discussion a little frustrating because it continues, Yglesias's words, the American habit of mixing up discussions of the price of houses with discussion of the price of land. (Though the habit is not restricted to Americans.) As he continues:
Land is a speculative commodity, comparable to a bond or a stock or oil futures. You might lose money buying Texas Hill Country land and you might make money. The house is more like a bulldozer or a camel. It’s expensive, and its potential resale value is a relevant issue when you’re considering a purchase, but it’s not an investment. Five years from now, your house, like your bulldozer, is going to be older, more broken, and cheaper than it is today. It’s the land the house/bulldozer is on that might be more expensive. That might be because it becomes more desirable as a place to live, or because climactic shifts make the lend better-suited for high-value agriculture, or because oil is found on the property, or because it acquires a better road connection, whatever. There are lots of perfectly ordinary reasons for land to go up or down in price. A house, by contrast, is a large decaying physical object.

So, is this land-with-housing-on-it in the Zoned Zone or in Flatland? (To use Krugman's apt terminology.) It makes a difference. Australia is all Zoned Zone. We have the most expensive land-with-houses in the Anglosphere (exempting Hong Kong, of course).

(How does such a low density country has such expensive housing land? Cue jokes about if the Soviet Union had taken over the Sahara in five years there would be a sand shortage. We took over the UK everything-need-official-approval land use system and our State Governments get lots of income from property taxes but none from income taxes--so restricting land supply increases their revenues and protects the land-values of incumbents, while many of our housing market entrants are not voters. BTW that revenue flow also screws with the incentive for State governments to provide infrastructure.)

Vancouver is Zoned Zone. Canada generally tends to be Flatland. California is Zoned Zone, Texas is Flatland. The UK is Zoned Zone, Germany is Flatland.

So, what do we predict for constrained-supply assets? Is it the same as supply-responsive assets? Perhaps not? The null hypothesis in Australia is, outside Tasmania, the real price of houses has a much lower likelihood of falling in (real) value than of rising. That is why two thirds of lending in Australia is on housing. All lending. We are a country highly leveraged on regulatory approval.

And I love the comment about evolution too :)

I agree that as we become more productive in everything else and devote less of our income to them, we invest more in housing and other investments. Investments and positional goods often act like collectors of whatever funds we have left. If population growth flattens or declines we could end up in a different world though, one where we individually expand to fill available space and the lowest value properties are abandoned once their maintenance exceeds their utility. Land is perhaps 15% of value in the countryside while it is over 50% in metropolises and can reach 90% or more in exclusive areas, so reductions in construction costs will only have limited effects on prices.

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