Persistence and Dispersion in the Housing Market

1. Motivation

House-price growth is persistent. When you regress current house-price growth on lagged house-price growth using monthly Zip-code level data,

(1) $\begin{align*} \Delta \bar{p}_{z,t} &= \alpha_z + \beta_z \cdot \Delta \bar{p}_{z,t-1} + \varepsilon_{z,t}, \end{align*}$

you find positive predictive power, $\langle \beta_z \rangle > 0$ . Using MSA- or county-level data gives similar results. If prices were growing faster than usual last month, then they’ll be growing faster than usual next month as well.

This persistence shouldn’t exist in a fully rational model with no frictions. Obviously, people aren’t necessarily fully rational and the housing market isn’t frictionless, but exactly which distortions are important for explaining the persistence in house-price growth? There’s been a lot of research trying to answer this question. People have “explained” the persistence in house-price growth using models with naive search/learning (Head, Lloyd-Ellis, and Sun (2013)), extrapolative beliefs (Glaeser and Nathanson (2015)), and lending frictions (Stein (1995) and Glaeser, Gottlieb, and Gyourko (2013)) to give just a few examples. By any reasonable count, there are now more answers than there are questions.

In a new working paper [link], Aurel Hizmo and I present a new stylized fact—namely, that it’s the places with the most homogeneous housing stock have the most persistent house-price growth—which helps discriminate between some of the many plausible explanations above. We find that Zip codes like 90713 (Lakewood, CA), where all the houses are really similar to one another, have very persistent house-price growth; whereas, Zip codes like 90402 (Santa Monica, CA), where the housing stock is really heterogeneous, have less persistent house-price growth. This fact suggests that naive search/learning, explanations which rely on house-to-house variation in quality, are unlikely to be key drivers of price-growth persistence. In this post, I outline a simple economic model to make this intuition more concrete: if naive search/learning is causing price-growth persistence, then it should be the places with the most heterogeneous housing stock that have the most persistent price growth.

2. House Values

Let’s study a Zip-code-level housing market with an aggregate supply of $\psi$ houses. Let $\bar{v}_T$ denote the average value of owning a home in this Zip code at time $T$ . Think about this as the consumption dividend from living in a randomly selected home in that Zip code. Suppose that this dividend is the sum of a whole bunch of independent Zip-code-level shocks,

(2) $\begin{align*} \bar{v}_T &= \bar{v}_0 + {\textstyle \sum_{t=1}^T} \, \bar{\epsilon}_t, \end{align*}$

with $\bar{\epsilon}_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0, \, \sigma_{\epsilon}^2)$ . For example, one month a park might get built (a positive shock), the next month a nearby grocery store might close (a negative shock), and so on… In the analysis below, I use the $\bar{x}$ notation to denote the Zip-code-level average of $x$ , where $x$ is any arbitrary variable in the model.

Now, suppose that within the Zip code there are $K$ different kinds of houses. For instance, if you’re thinking about Lakewood, CA where all the houses are identical, then $K=1$ . By contrast, if you’re thinking about Santa Monica, CA where the housing stock is very heterogeneous, then $K \gg 1$ . Each aggregate Zip-code-level shock is the average of the shocks to each of the $K$ housing types,

(3) $\begin{align*} \bar{\epsilon}_t &= {\textstyle \frac{1}{K} \cdot \sum_{k=1}^K} \, \epsilon_{k,t}, \end{align*}$

with $\epsilon_{k,t} \sim \mathrm{N}(0, \, K \cdot \sigma_{\epsilon}^2)$ . For example, the grocery-storing-closing shock might be a really negative shock for the part of the Zip code closest to the store, but only a minor inconvenience for the part of the Zip code farthest from the store who only shopped there occasionally.

3. Information Structure

Next, let’s turn our attention to the information structure in the model. We want to capture the idea that it’s more difficult to learn about how awesome it would be to live in a Zip code if the housing stock is more heterogeneous. After all, if you’ve seen one house in Lakewood, CA, then you’ve seen them all. By contrast, the first house you see in Santa Monica, CA might not be very representative of what it’s like to live there since all the houses are so different.

Here’s how the information structure works. There’s a unit mass of buyers and that each buyer can only view one kind of house each period. At time $t$ , buyers begin seeing signals about, $\bar{\epsilon}_{[t-1]+K}$ , that is, the Zip-code-level shock that will occur at time $([t-1]+K)$ . But, buyers don’t get to see the full shock right away. A randomly selected group of $\sfrac{1}{K}$ buyers will see the signal for the first kind of house, $\epsilon_{1,[t-1]+K}$ , another randomly selected group of $\sfrac{1}{K}$ buyers will see the signal for the second kind of house, $\epsilon_{2,[t-1]+K}$ , and so on… until the $K$ th group of $\sfrac{1}{K}$ buyers sees the signal for the $K$ th kind of house, $\epsilon_{K,[t-1]+K}$ . So, each of the signals about the Zip code’s time $([t-1]+K)$ shock will been seen by $(\sfrac{1}{K})$ th of the buyers at time $t$ . You can think about this as buyer shopping around and each seeing different kinds of houses. After seeing their initial signals, each buyer can then form posterior beliefs about the value of owning a house in that Zip code.

At time $(t+1)$ , the same process then repeats with each buyer viewing the time $([t-1]+K)$ signal for a different kind of house. A randomly selected group of $\sfrac{1}{K}$ buyers who haven’t already looked at the first kind of house will see the signal for the first kind of house, $\epsilon_{1,[t-1]+K}$ , another randomly selected group of $\sfrac{1}{K}$ buyers who haven’t already looked at the second kind of house will see the signal for the second kind of house, $\epsilon_{2,[t-1]+K}$ , and so on… until a $K$ th group of $\sfrac{1}{K}$ buyers that hasn’t already looked at the $K$ th kind of house sees the signal for the $K$ th kind of houses, $\epsilon_{K,[t-1]+K}$ . So, each of the signals about the Zip code’s time $([t-1]+K)$ shock will been seen by $(\sfrac{2}{K})$ th of the buyers at time $(t+1)$ . This learning process keeps on going until all buyers have seen all signals for the time $([t-1]+K)$ Zip-code-level shock by the end of period $([t-1]+K)$ .

This information structure is analytically convenient because it makes sure that each buyer has seen the same amount of information at each point in time, even though no $2$ buyers will have seen the same signals. So, the parameter $K$ can be thought of as a proxy for information flow or the rate of learning. Larger values of $K$ mean that it takes buyers longer to learn about the Zip code’s fundamental value. Note that this setup is formally equivalent to the “rotation” convention used in Hong and Stein (1999).

4. Buyer Preferences

Finally, to complete the model, let’s have a look at the home-buyers’ preferences. I assume that buyers have constant-absolute–risk-aversion (CARA) utility with risk-aversion parameter, $\rho > 0$ . Each period buyers choose how many shares of housing to buy in the Zip code, $x_{b,t}$ , and how many shares of the riskless asset to buy, $y_{b,t}$ , in order to maximize their time $T$ wealth, $w_{b,T}$ ,

(4) $\begin{align*} \max_{(x_{b,t}, \, y_{b,t})_{t=1}^T} \, \mathrm{E}_{b,t}[ \, - \exp\{ \, - \rho \cdot w_{b,T} \, \} \, ] \quad \text{s.t.} \quad w_0 \geq \bar{p}_t \cdot x_{b,t} + y_{b,t}. \end{align*}$

I normalize the net riskless rate is $r_f = 0$ .

For simplicity, let’s assume that at every time, $t$ , the buyers formulate their housing demand based on the static-optimization notion that they’re going to buy and hold until time $T$ . This’ll make the pricing equations really clean and won’t affect the intuition. However, to be clear, it does introduce an element of time inconsistency. Although home buyers formulate their time $t$ demand based on the assumption that they won’t re-trade, they violate this assumption if they are active in later periods.

5. Fully-Rational Equilibrium

With the model in place, we can now turn our attention to the solution. First, let’s study the fully-rational equilibrium as a benchmark case. If each buyer recognizes that the market-clearing price reveals information about the set of signals that the other buyers got, then, by the logic of Grossman (1976), prices should be fully revealing because there is no uncertainty about housing supply,

(5) $\begin{align*} \bar{p}_t^{\text{rational}} &= \bar{v}_{[t-1]+K} - \mathrm{f}(\rho) \cdot \psi \\ &= \bar{v}_{t-1} + {\textstyle \sum_{k=0}^{K-1}} \bar{\epsilon}_{t+k} - \mathrm{f}(\rho) \cdot \psi, \end{align*}$

where $\mathrm{f}(\rho)$ is a function of the home buyers’ risk-aversion parameter. This is why you need the random-asset-supply assumption in models like Grossman and Stiglitz (1980) and Hellwig (1980).

Obviously, in this setup there isn’t going to be any price-growth persistence,

(6) $\begin{align*} \mathrm{Cov}[\Delta \bar{p}_t^{\text{rational}}, \, \Delta \bar{p}_{t-1}^{\text{rational}}] = 0, \end{align*}$

because each of the Zip-code-level shocks is drawn independently. This result shows that search/learning on its own is not enough to generate persistence. You need people to be naive as well. You need people to be ignoring some information.

6. Equilibrium with Persistence

Suppose buyers don’t condition on current or past prices. e.g., think about this as an extreme form of a cursed equilibrium as in Eyster, Rabin, and Vayanos (2015) or just as a Walrasian equilibrium with private valuations. In this setting, the Zip-code level house prices will be given by

(7) $\begin{align*} \bar{p}_t^{\text{na\"{i}ve}} &= \bar{v}_{t-1} + {\textstyle \sum_{k=0}^{K-1}} \left({\textstyle \frac{K - k}{K}}\right) \cdot \bar{\epsilon}_{t+k} - \mathrm{f}(\rho) \cdot \psi, \end{align*}$

where again $\mathrm{f}(\rho)$ is a function of buyers’ risk-aversion parameter. Because buyers ignore some information, it takes longer for shocks to fundamentals to work their way into prices. As a result, price movements are now persistent.

Let’s compute the level of persistence to see how it varies with the heterogeneity of the housing stock, $K$ . First, note that changes in the Zip-code-level price are given by:

(8) $\begin{align*} \Delta \bar{p}_t^{\text{na\"{i}ve}} &= {\textstyle \frac{1}{K}} \cdot {\textstyle \sum_{k=0}^{K-1}} \bar{\epsilon}_{t+k}. \end{align*}$

Thus, the level of auto-correlation is given by:

(9) $\begin{align*} \frac{ \mathrm{Cov}[\Delta \bar{p}_t^{\text{na\"{i}ve}}, \, \Delta \bar{p}_{t-1}^{\text{na\"{i}ve}}] }{ \mathrm{Var}[\Delta \bar{p}_{t-1}^{\text{na\"{i}ve}}] } &= \frac{K-1}{K} \end{align*}$

With this formulation, you see that the more heterogeneous the housing stock (i.e., the higher the $K$ ), the more persistent the house-price level will be. After all, house prices are a weighted average of past shocks in this setting. Buyers consistently under-react to new information.

You can write down this sort of naive search/learning model in lots of ways. But, no matter how you do it, you always get this sort of result. When assets are more heterogeneous—that is, when there is more to learn about—price growth is more persistent. Thus, when Aurel and I find that Zip codes with the most homogeneous housing stock have the most price-growth persistence, this is strong evidence that naive search/learning isn’t the explanation. It’s got to be something else like, for example, extrapolative beliefs.